Found problems: 55
MMPC Part II 1996 - 2019, 2004
[b]p1.[/b] The following figure represents a rectangular piece of paper $ABCD$ whose dimensions are $4$ inches by $3$ inches. When the paper is folded along the line segment $EF$, the corners $A$ and $C$ coincide.
(a) Find the length of segment $EF$.
(b) Extend $AD$ and $EF$ so they meet at $G$. Find the area of the triangle $\vartriangle AEG$.
[img]https://cdn.artofproblemsolving.com/attachments/d/4/e8844fd37b3b8163f62fcda1300c8d63221f51.png[/img]
[b]p2.[/b] (a) Let $p$ be a prime number. If $a, b, c$, and $d$ are distinct integers such that the equation $(x -a)(x - b)(x - c)(x - d) - p^2 = 0$ has an integer solution $r$, show that $(r - a) + (r - b) + (r - c) + (r - d) = 0$.
(b) Show that $r$ must be a double root of the equation $(x - a)(x - b)(x - c)(x - d) - p^2 = 0$.
[b]p3.[/b] If $\sin x + \sin y + \sin z = 0$ and $\cos x + \cos y + \cos z = 0$, prove the following statements.
(a) $\cos (x - y) = -\frac12$
(b) $\cos (\theta - x) + \cos(\theta - y) + \cos (\theta - z) = 0$, for any angle $\theta$.
(c) $\sin^2 x + \sin^2 y + \sin^2 z =\frac32$
[b]p4.[/b] Let $|A|$ denote the number of elements in the set $A$.
(a) Construct an infinite collection $\{A_i\}$ of infinite subsets of the set of natural numbers such that $|A_i \cap A_j | = 0$ for $i \ne j$.
(b) Construct an infinite collection $\{B_i\}$ of infinite subsets of the set of natural numbers such that $|B_i \cap B_j |$ gives a distinct integer for every pair of $i$ and $j$, $i \ne j$.
[b]p5.[/b] Consider the equation $x^4 + y^4 = z^5$.
(a) Show that the equation has a solution where $x, y$, and $z$ are positive integers.
(b) Show that the equation has infinitely many solutions where $x, y$, and $z$ are positive integers.
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MMPC Part II 1958 - 95, 1978
[b]p1.[/b] A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. (Justify your answer.)
[b]p2.[/b] The residents of Andromeda use only bills of denominations $\$3 $and $\$5$ . All payments are made exactly, with no change given. What whole-dollar payments are not possible? (Justify your answer.)
[b]p3.[/b] A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$ . Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal fotal weights. Find all possible weights for the objects. (Justify your answer.)
[b]p4.[/b] Let $P(x) = x^3 + x^2 - 1$ and $Q(x) = x^3 - x - 1$ . Given that $r$ and $s$ are two distinct solutions of $P(x) = 0$ , prove that $rs$ is a solution of $Q(x) = 0$
[b]p5.[/b] Given: $\vartriangle ABC$ with points $A_1$ and $A_2$ on $BC$ , $B_1$ and $B_2$ on $CA$, and $C_1$ and $C_2$ on $AB$.
$A_1 , A_2, B_1 , B_2$ are on a circle,
$B_1 , B_2, C_1 , C_2$ are on a circle, and
$C_1 , C_2, A_1 , A_2$ are on a circle.
The centers of these circles lie in the interior of the triangle.
Prove: All six points $A_1$ , $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are on a circle.
[img]https://cdn.artofproblemsolving.com/attachments/7/2/2b99ddf4f258232c910c062e4190d8617af6fa.png[/img]
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MMPC Part II 1996 - 2019, 2013
[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.
[b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$.
[img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img]
[b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color.
[b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points.
(b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property.
[b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$.
(a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$.
(b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$.
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MMPC Part II 1958 - 95, 1995
[b]p1.[/b] (a) Brian has a big job to do that will take him two hours to complete. He has six friends who can help him. They all work at the same rate, somewhat slower than Brian. All seven working together can finish the job in $45$ minutes. How long will it take to do the job if Brian worked with only three of his friends?
(b) Brian could do his next job in $N$ hours, working alone. This time he has an unlimited list of friends who can help him, but as he moves down the list, each friend works more slowly than those above on the list. The first friend would take $kN$ ($k > 1$) hours to do the job alone, the second friend would take $k^2N$ hours alone, the third friend would take $k^3N$ hours alone, etc. Theoretically, if Brian could get all his infinite number of friends to help him, how long would it take to complete the job?
[b]p2.[/b] (a) The centers of two circles of radius $1$ are two opposite vertices of a square of side $1$. Find the area of the intersection of the two circles.
(b) The centers of two circles of radius $1$ are two consecutive vertices of a square of side $1$. Find the area of the intersection of the two circles and the square.
(c) The centers of four circles of radius $1$ are the vertices of a square of side $1$. Find the area of the intersection of the four circles.
[b]p3.[/b] For any real number$ x$, $[x]$ denotes the greatest integer that does not exceed $x$. For example, $[7.3] = 7$, $[10/3] = 3$, $[5] = 5$. Given natural number $N$, denote as $f(N)$ the following sum of $N$ integers:
$$f(N) = [N/1] + [N/2] + [N/3] + ... + [N/n].$$
(a) Evaluate $f(7) - f(6)$.
(b) Evaluate $f(35) - f(34)$.
(c) Evaluate (with explanation) $f(1996) - f(1995)$.
[b]p4.[/b] We will say that triangle $ABC$ is good if it satisfies the following conditions: $AB = 7$, the other two sides are integers, and $\cos A =\frac27$.
(a) Find the sides of a good isosceles triangle.
(b) Find the sides of a good scalene (i.e. non-isosceles) triangle.
(c) Find the sides of a good scalene triangle other than the one you found in (b) and prove that any good triangle is congruent to one of the three triangles you have found.
[b]p5.[/b] (a) A bag contains nine balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. How many of the nine balls were of one color and how many of the other color?
(b) A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Find the exact value of $N$ and determine how many of the $N$ balls were of one color and how many of the other color.
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MMPC Part II 1958 - 95, 1975
[b]p1.[/b] a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this.
b) Repeat part a) with "five" replacing "four" throughout.
[b]p2.[/b] Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer.
[b]p3. [/b]Let $S$ denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is $2S$, and that df the cubes is $64S/13$. Find the first three terms of the original series.
[b]p4.[/b] $A$, $B$ and $C$ are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line $\ell$, the sum of the distances of the points $A, B$, and $C$ above line $\ell$ is constant.
[img]https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png[/img]
[b]p5.[/b] A set of $n$ numbers $x_1,x_2,x_3,...,x_n$ (where $n>1$) has the property that the $k^{th}$ number (that is, $x_k$ ) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript o $x_k$ ), and this is true for each $k = 1,2,3,...,n$.
a) SoIve for these $n$ numbers
b) Find whether at least one of these $n$ numbers can be an integer.
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MMPC Part II 1958 - 95, 1962
[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular.
Decide whether this statement is a true or false proposition in euclidean geometry.
If it is true, prove it; if false, produce a counterexample.
[b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$.
[b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance?
[b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer?
[b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|}
\hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline \end{tabular}$
i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order.
ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row.
Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order.
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MMPC Part II 1958 - 95, 1966
[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches.
[b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$.
[img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img]
[b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$
[b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute.
[b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties:
(i) each is less than the sum of the other three, and
(ii) each is a factor of the sum of the other three.
Prove that at least two of the numbers must be equal.
(An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.)
[b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties:
(i) The two triangles have no common vertex.
(ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other.
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MMPC Part II 1996 - 2019, 2001
[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction.
Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible.
[b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$.
[b]p3.[/b] Consider the following two questions:
Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY."
Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates?
a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$.
b. Answer either Question $1$ or Question $2$.
[b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions:
$\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$
$\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$
a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$.
b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin
((2n +1)x)$$
[b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$.
a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges.
b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point.
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MMPC Part II 1958 - 95, 1982
[b]p1.[/b] Sarah needed a ride home to the farm from town. She telephoned for her father to come and get her with the pickup truck. Being eager to get home, she began walking toward the farm as soon as she hung up the phone. However, her father had to finish milking the cows, so could not leave to get her until fifteen minutes after she called. He drove rapidly to make up for lost time.
They met on the road, turned right around and drove back to the farm at two-thirds of the speed her father drove coming. They got to the farm two hours after she had called. She walked and he drove both ways at constant rates of speed.
How many minutes did she spend walking?
[b]p2.[/b] Let $A = (a,b)$ be any point in a coordinate plane distinct from the origin $O$. Let $M$ be the midpoint of $OA$, and let $P$ be a point such that $MP$ is perpendicular to $OA$ and the lengths $\overline{MP}$ and $\overline{OM}$ are equal. Determine the coordinates $(x,y)$ of $P$ in terms of $a$ and $b$. Give all possible solutions.
[b]p3.[/b] Determine the exact sum of the series
$$\frac{1}{1 \cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \frac{1}{3\cdot 4\cdot 5} + ... + \frac{1}{98\cdot 99\cdot 100}$$
[b]p4.[/b] A six pound weight is attached to a four foot nylon cord that is looped over two pegs in the manner shown in the drawing. At $B$ the cord passes through a small loop in its end. The two pegs $A$ and $C$ are one foot apart and are on the same level. When the weight is released the system obtains an equilibrium position. Determine angle $ABC$ for this equilibrium position, and verify your answer. (Your verification should assume that friction and the weight of the cord are both negligible, and that the tension throughout the cord is a constant six pounds.)
[img]https://cdn.artofproblemsolving.com/attachments/a/1/620c59e678185f01ca8743c39423234d5ba04d.png[/img]
[b]p5.[/b] The four corners of a rectangle have the property that when they are taken three at a time, they determine triangles all of which have the same perimeter. We will consider whether a set of five points can have this property.
Let $S = \{p_1, p_2, p_3, p_4, p_5\}$ be a set of five points. For each $i$ and $j$, let $d_{ij}$ denote the distance from $p_i$ to $p_j$. Suppose that $S$ has the property that all triangles with vertices in $S$ have the same perimeter.
(a) Prove that $d$ must be the same for every pair $(i,j)$ with $i \ne j$.
(b) Can such a five-element set be found in three dimensional space? Justify your answer.
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MMPC Part II 1996 - 2019, 2017
[b]p1.[/b] Consider a normal $8 \times 8$ chessboard, where each square is labelled with either $1$ or $-1$. Let $a_k$ be the product of the numbers in the $k$th row, and let $b_k$ be the product of the numbers in the $k$th column. Find, with proof, all possible values of $\sum^8_{k=1}(a_kb_k)$.
[b]p2.[/b] Let $\overline{AB}$ be a line segment with $AB = 1$, and $P$ be a point on $\overline{AB}$ with $AP = x$, for some $0 < x < 1$. Draw circles $C_1$ and $C_2$ with $\overline{AP}$, $\overline{PB}$ as diameters, respectively. Let $\overline{AB_1}$, $\overline{AB_2}$ be tangent to $C_2$ at $B_1$ and $B_2$, and let $\overline{BA_1}$;$\overline{BA_2}$ be tangent to $C_1$ at $A_1$ and $A_2$. Now $C_3$ is a circle tangent to $C_2$, $\overline{AB_1}$, and $\overline{AB_2}$; $C_4$ is a circle tangent to $C_1$, $\overline{BA_1}$, and $\overline{BA_2}$.
(a) Express the radius of $C_3$ as a function of $x$.
(b) Prove that $C_3$ and $C_4$ are congruent.
[img]https://cdn.artofproblemsolving.com/attachments/c/a/fd28ad91ed0a4893608b92f5ccbd01088ae424.png[/img]
[b]p3.[/b] Suppose that the graphs of $y = (x + a)^2$ and $x = (y + a)^2$ are tangent to one another at a point on the line $y = x$. Find all possible values of $a$.
[b]p4.[/b] You may assume without proof or justification that the infinite radical expressions $\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a-...}}}}$ and $\sqrt{a-\sqrt{a+\sqrt{a-\sqrt{a+...}}}}$ represent unique values for $a > 2$.
(a) Find a real number $a$ such that $$\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a+...}}}}= 2017$$
(b) Show that
$$\sqrt{2018-\sqrt{2018+\sqrt{2018-\sqrt{2018+...}}}}=\sqrt{2017-\sqrt{2017-\sqrt{2017-\sqrt{2017-...}}}}$$
[b]p5.[/b] (a) Suppose that $m, n$ are positive integers such that $7n^2 - m^2 > 0$. Prove that, in fact, $7n^2 - m^2 \ge 3$.
(b) Suppose that $m, n$ are positive integers such that $\frac{m}{n} <\sqrt7$. Prove that, in fact, $\frac{m}{n}+\frac{1}{mn}
<\sqrt7$.
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MMPC Part II 1996 - 2019, 1997
[b]p1.[/b] It can be shown in Calculus that the area between the x-axis and the parabola $y=kx^2$ (к is a positive constant) on the $x$-interval $0 \le x \le a$ is $\frac{ka^3}{3}$
a) Find the area between the parabola $y=4x^2$ and the x-axis for $0 \le x \le 3$.
b) Find the area between the parabola $y=5x^2$ and the x-axis for $-2 \le x \le 4$.
c) A square $2$ by $2$ dartboard is situated in the $xy$-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point $(0, 1)$ than to the bottom edge of the dartboard.
[b]p2.[/b] When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. For example, given the determinant $\begin{vmatrix} 1 & 2 & 3 \\
2 & 6 & 2 \\
3 & 10 & 4
\end{vmatrix}$ we add $-2$ times the first row to the second row, then add $-2$ times the second row to the third row, giving the new determinant $\begin{vmatrix} 1 & 2 & 3 \\
0 & 2 & -4 \\
0 & 0 & 3
\end{vmatrix}$ , and the value is the product of the diagonal entries: $6$.
a) Transform this determinant into another determinant with zeroes everywhere below the main diagonal, and find its value: $\begin{vmatrix} 1 & 3 & -1 \\
4 & 7 & 2 \\
3 & -6 & 5
\end{vmatrix}$
b) Do the same for this determinant: $\begin{vmatrix} 0 & 1 & 2 & 3 \\
1 & 0 & 1 & 2 \\
2 & 1 & 0 & 1 \\
3 & 2 & 1 & 0
\end{vmatrix}$
[b]p3.[/b] In Pascal’s triangle, the entries at the ends of each row are both $1$, and otherwise each entry is the sum of the two entries diagonally above it:
Row Number
$0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$
$1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\,\,1$
$2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\, 2 \,\,1$
$3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1\,\, 3 \,\, 3 \,\, 1$
$4\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \,\,4 \,\, 6 \,\, 4 \,\, 1$
$...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...$
This triangle gives the binomial coefficients in expansions like $( a + b)^3 = 1a^3 + 3a^2 b + 3 ab^2 + 1b^3$ .
a) What is the sum of the numbers in row #$5$ of Pascal's triangle?
b) What is the sum of the numbers in row #$n$ of Pascal's triangle?
c) Show that in row #$6$ of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row.
d) Prove that in row #$n$ of Pascal's triangle, the sum of ail the numbers is exactly twice the sum of the numbers in the odd positions of that row.
[b]p4.[/b] The product: of several terms is sometimes described using the symbol $\Pi$ which is capital pi, the Greek equivalent of $p$, for the word "product". For example the symbol $\prod^4_{k=1}(2k +1)$ means the product of numbers of the form $(2k + 1)$, for $k=1,2,3,4$. Thus it equals $945$.
a) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k}{k + 2}$
b) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k^2 + 10k+ 17}{k^2+4k + 41}$
c) Evaluate as a reduced fraction $\prod_{k=1}^{\infty}\frac{k^3-1}{k^3+1}$
[b]p5.[/b] a) In right triangle $CAB$, the median $AF$, the angle bisector $AE$, and the altitude $AD$ divide the right angld $A$ into four equal angles. If $AB = 1$, find the area of triangle $AFE$.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/0d4a83e58a65c2546ce25d1081b99d45e30729.png[/img]
b) If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2000
[b]p1.[/b] Jose,, Luciano, and Placido enjoy playing cards after their performances, and you are invited to deal. They use just nine cards, numbered from $2$ through $10$, and each player is to receive three cards. You hope to hand out the cards so that the following three conditions hold:
A) When Jose and Luciano pick cards randomly from their piles, Luciano most often picks a card higher than Jose,
B) When Luciano and Placido pick cards randomly from their piles, Placido most often picks a card higher than Luciano,
C) When Placido and Jose pick cards randomly from their piles, Jose most often picks a card higher than Placido.
Explain why it is impossible to distribute the nine cards so as to satisfy these three conditions, or give an example of one such distribution.
[b]p2.[/b] Is it possible to fill a rectangular box with a finite number of solid cubes (two or more), each with a different edge length? Justify your answer.
[b]p3.[/b] Two parallel lines pass through the points $(0, 1)$ and $(-1, 0)$. Two other lines are drawn through $(1, 0)$ and $(0, 0)$, each perpendicular to the ¯rst two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines.
[b]p4.[/b] Suppose $a_1, a_2, a_3,...$ is a sequence of integers that represent data to be transmitted across a communication channel. Engineers use the quantity
$$G(n) =(1 - \sqrt3)a_n -(3 - \sqrt3)a_{n+1} +(3 + \sqrt3)a_{n+2}-(1+ \sqrt3)a_{n+3}$$
to detect noise in the signal.
a. Show that if the numbers $a_1, a_2, a_3,...$ are in arithmetic progression, then $G(n) = 0$ for all $n = 1, 2, 3, ...$.
b. Show that if $G(n) = 0$ for all $n = 1, 2, 3, ...$, then $a_1, a_2, a_3,...$ is an arithmetic progression.
[b]p5.[/b] The Olive View Airline in the remote country of Kuklafrania has decided to use the following rule to establish its air routes: If $A$ and $B$ are two distinct cities, then there is to be an air route connecting $A$ with $B$ either if there is no city closer to $A$ than $B$ or if there is no city closer to $B$ than $A$. No further routes will be permitted. Distances between Kuklafranian cities are never equal. Prove that no city will be connected by air routes to more than ¯ve other cities.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2018
[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below.
Let $r(x)$ be the ratio of the area of region I to the area of region II.
(i) Find $r(1)$. Justify your answer.
(ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img]
[b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party.
(i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$.
(ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ .
[b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$.
(i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer.
(ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer.
[b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.)
(a) Determine with proof the value of $f(120)$.
(b) Determine with proof the smallest value $x$ for which $f(x) = 8$.
(c) Show that the range of this function is the set of all positive integers.
[b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2019
[b]p1.[/b] Consider a parallelogram $ABCD$ with sides of length $a$ and $b$, where $a \ne b$. The four points of intersection of the bisectors of the interior angles of the parallelogram form a rectangle $EFGH$. A possible configuration is given below.
Show that $$\frac{Area(ABCD)}{Area(EFGH)}=\frac{2ab}{(a - b)^2}$$
[img]https://cdn.artofproblemsolving.com/attachments/e/a/afaf345f2ef7c8ecf4388918756f0b56ff20ef.png[/img]
[b]p2.[/b] A metal wire of length $4\ell$ inches (where $\ell$ is a positive integer) is used as edges to make a cardboard rectangular box with surface area $32$ square inches and volume $8$ cubic inches. Suppose that the whole wire is used.
(i) Find the dimension of the box if $\ell= 9$, i.e., find the length, the width, and the height of the box without distinguishing the different orders of the numbers. Justify your answer.
(ii) Show that it is impossible to construct such a box if $\ell = 10$.
[b]p3.[/b] A Pythagorean n-tuple is an ordered collection of counting numbers $(x_1, x_2,..., x_{n-1}, x_n)$ satisfying the equation $$x^2_1+ x^2_2+ ...+ x^2_{n-1} = x^2_{n}.$$
For example, $(3, 4, 5)$ is an ordinary Pythagorean $3$-tuple (triple) and $(1, 2, 2, 3)$ is a Pythagorean $4$-tuple.
(a) Given a Pythagorean triple $(a, b, c)$ show that the $4$-tuple $(a^2, ab, bc, c^2)$ is Pythagorean.
(b) Extending part (a) or using any other method, come up with a procedure that generates Pythagorean $5$-tuples from Pythagorean $3$- and/or $4$-tuples. Few numerical examples will not suffice. You have to find a method that will generate infinitely many such $5$-tuples.
(c) Find a procedure to generate Pythagorean $6$-tuples from Pythagorean $3$- and/or $4$- and/or $5$-tuples.
Note. You can assume without proof that there are infinitely many Pythagorean triples.
[b]p4.[/b] Consider the recursive sequence defined by $x_1 = a$, $x_2 = b$ and $$x_{n+2} =\frac{x_{n+1} + x_n - 1}{x_n - 1}, n \ge 1 .$$
We call the pair $(a, b)$ the seed for this sequence. If both $a$ and $b$ are integers, we will call it an integer seed.
(a) Start with the integer seed $(2, 2019)$ and find $x_7$.
(b) Show that there are infinitely many integer seeds for which $x_{2020} = 2020$.
(c) Show that there are no integer seeds for which $x_{2019} = 2019$.
[b]p5.[/b] Suppose there are eight people at a party. Each person has a certain amount of money. The eight people decide to play a game. Let $A_i$, for $i = 1$ to $8$, be the amount of money person $i$ has in his/her pocket at the beginning of the game. A computer picks a person at random. The chosen person is eliminated from the game and their money is put into a pot. Also magically the amount of money in the pockets of the remaining players goes up by the dollar amount in the chosen person's pocket. We continue this process and at the end of the seventh stage emerges a single person and a pot containing $M$ dollars. What is the expected value of $M$? The remaining player gets the pot and the money in his/her pocket. What is the expected value of what he/she takes home?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1992
[b]p1.[/b] The English alphabet consists of $21$ consonants and $5$ vowels. (We count $y$ as a consonant.)
(a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be $4$ consecutive consonants.
(b) Give a list to show that there need not be $5$ consecutive consonants.
(c) Suppose that all the letters are arranged in a circle. Prove that there must be $5$ consecutive consonants.
[b]p2.[/b] From the set $\{1,2,3,... , n\}$, $k$ distinct integers are selected at random and arranged in numerical order (lowest to highest). Let $P(i, r, k, n)$ denote the probability that integer $i$ is in position $r$. For example, observe that $P(1, 2, k, n) = 0$.
(a) Compute $P(2, 1,6,10)$.
(b) Find a general formula for $P(i, r, k, n)$.
[b]p3.[/b] (a) Write down a fourth degree polynomial $P(x)$ such that $P(1) = P(-1)$ but $P(2) \ne P(-2)$
(b) Write down a fifth degree polynomial $Q(x)$ such that $Q(1) = Q(-1)$ and $Q(2) = Q(-2)$ but $Q(3) \ne Q(-3)$.
(c) Prove that, if a sixth degree polynomial $R(x)$ satisfies $R(1) = R(-1)$, $R(2) = R(-2)$, and $R(3) = R(-3)$, then $R(x) = R(-x)$ for all $x$.
[b]p4.[/b] Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number $N$ of distinct values among these sums.
(a) Give an example of five numbers yielding the smallest possible value of $N$. What is this value?
(b) Give an example of five numbers yielding the largest possible value of $N$. What is this value?
(c) Prove that the values of $N$ you obtained in (a) and (b) are the smallest and largest possible ones.
[b]p5.[/b] Let $A_1A_2A_3$ be a triangle which is not a right triangle. Prove that there exist circles $C_1$, $C_2$, and $C_3$ such that $C_2$ is tangent to $C_3$ at $A_1$, $C_3$ is tangent to $C_1$ at $A_2$, and $C_1$ is tangent to $C_2$ at $A_3$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2008
[b]p1.[/b] Compute $$\left(\frac{1}{10}\right)^{\frac12}\left(\frac{1}{10^2}\right)^{\frac{1}{2^4}}\left(\frac{1}{10^3}\right)^{\frac{1}{2^3}} ...$$
[b]p2.[/b] Consider the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...,$ where the positive integer $m$ appears $m$ times. Let $d(n)$ denote the $n$th element of this sequence starting with $n = 1$. Find a closed-form formula for $d(n)$.
[b]p3.[/b] Let $0 < \theta < \frac{\pi}{2}$, prove that $$ \left( \frac{\sin^2 \theta}{2}+\frac{2}{\cos^2 \theta} \right)^{\frac14}+ \left( \frac{\cos^2 \theta}{2}+\frac{2}{\sin^2 \theta} \right)^{\frac14} \ge (68)^{\frac14} $$ and determine the value of \theta when the inequality holds as equality.
[b]p4.[/b] In $\vartriangle ABC$, parallel lines to $AB$ and $AC$ are drawn from a point $Q$ lying on side $BC$. If $a$ is used to represent the ratio of the area of parallelogram $ADQE$ to the area of the triangle $\vartriangle ABC$,
(i) find the maximum value of $a$.
(ii) find the ratio $\frac{BQ}{QC}$ when $a =\frac{24}{49}.$
[img]https://cdn.artofproblemsolving.com/attachments/5/8/eaa58df0d55e6e648855425e581a6ba0ad3ea6.png[/img]
[b]p5.[/b] Prove the following inequality
$$\frac{1}{2009} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8}...\frac{2007}{2008}<\frac{1}{40}$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].Thanks to gauss202 for sending the problems.
MMPC Part II 1958 - 95, 1981
[b]p1.[/b] A canoeist is paddling upstream in a river when she passes a log floating downstream,, She continues upstream for awhile, paddling at a constant rate. She then turns around and goes downstream and paddles twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream?
[b]p2.[/b] Let $g(x) =1-\frac{1}{x}$ and define $g_1(x) = g(x)$ and $g_{n+1}(x) = g(g_n(x))$ for $n = 1,2,3, ...$. Evaluate $g_3(3)$ and $g_{1982}(l982)$.
[b]p3.[/b] Let $Q$ denote quadrilateral $ABCD$ where diagonals $AC$ and $BD$ intersect. If each diagonal bisects the area of $Q$ prove that $Q$ must be a parallelogram.
[b]p4.[/b] Given that: $a_1, a_2, ..., a_7$ and $b_1, b_2, ..., b_7$ are two arrangements of the same seven integers, prove that the product $(a_1-b_1)(a_2-b_2)...(a_7-b_7)$ is always even.
[b]p5.[/b] In analyzing the pecking order in a finite flock of chickens we observe that for any two chickens exactly one pecks the other. We decide to call chicken $K$ a king provided that for any other chicken $X, K$ necks $X$ or $K$ pecks a third chicken $Y$ who in turn pecks $X$. Prove that every such flock of chickens has at least one king. Must the king be unique?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2010
[b]p1.[/b] Let $x_1 = 0$, $x_2 = 1/2$ and for $n >2$, let $x_n$ be the average of $x_{n-1}$ and $x_{n-2}$. Find a formula for $a_n = x_{n+1} - x_{n}$, $n = 1, 2, 3, \dots$. Justify your answer.
[b]p2.[/b] Given a triangle $ABC$. Let $h_a, h_b, h_c$ be the altitudes to its sides $a, b, c,$ respectively. Prove: $\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}$ Is it possible to construct a triangle with altitudes $7$, $11$, and $20$? Justify your answer.
[b]p3.[/b] Does there exist a polynomial $P(x)$ with integer coefficients such that $P(0) = 1$, $P(2) = 3$ and $P(4) = 9$? Justify your answer.
[b]p4.[/b] Prove that if $\cos \theta$ is rational and $n$ is an integer, then $\cos n\theta$ is rational. Let $\alpha=\frac{1}{2010}$. Is $\cos \alpha $ rational ? Justify your answer.
[b]p5.[/b] Let function $f(x)$ be defined as $f(x) = x^2 + bx + c$, where $b, c$ are real numbers.
(A) Evaluate $f(1) -2f(5) + f(9)$ .
(B) Determine all pairs $(b, c)$ such that $|f(x)| \le 8$ for all $x$ in the interval $[1, 9]$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2003
[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$.
(a) Show that if the equation has a solution, then $n$ is even.
(b) Suppose $n$ is divisible by $4$. Show that the equation has a solution.
(c) Show that if the equation has a solution, then $n$ is divisible by $4$.
[b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$.
(b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$.
[b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$.
(b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$
(c) Determine the exact value of the following infinite sum.
$$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$
[b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$.
(b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete.
[b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure).
As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$.
[img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2014
[b]p1.[/b] If $P$ is a (convex) polygon, a triangulation of $P$ is a set of line segments joining pairs of corners of $P$ in such a way that $P$ is divided into non-overlapping triangles, each of which has its corners at corners of $P$. For example, the following are different triangulations of a square.
(a) Prove that if $P$ is an $n$-gon with $n > 3$, then every triangulation of $P$ produces at least two triangles $T_1$, $T_2$ such that two of the sides of $T_i$, $i = 1$ or $2$ are also sides of $P$.
(b) Find the number of different possible triangulations of a regular hexagon.
[img]https://cdn.artofproblemsolving.com/attachments/9/d/0f760b0869fafc882f293846c05d182109fb78.png[/img]
[b]p2.[/b] There are $n$ students, $n \ge 2$, and $n + 1$ cubical cakes of volume $1$. They have the use of a knife. In order to divide the cakes equitably they make cuts with the knife. Each cut divides a cake (or a piece of a cake) into two pieces.
(a) Show that it is possible to provide each student with a volume $(n + 1)/n$ of a cake while making no more than $n - 1$ cuts.
(b) Show that for each integer $k$ with $2 \le k \le n$ it is possible to make $n - 1$ cuts in such a way that exactly $k$ of the $n$ students receive an entire (uncut) cake in their portion.
[b]p3. [/b]The vertical lines at $x = 0$, $x = \frac12$ , $x = 1$, $x = \frac32$ ,$...$ and the horizontal lines at $y = 0$, $y = \frac12$ , $y = 1$, $y = \frac32$ ,$ ...$ subdivide the first quadrant of the plane into $\frac12 \times \frac12$ square regions. Color these regions in a checkerboard fashion starting with a black region near the origin and alternating black and white both horizontally and vertically.
(a) Let $T$ be a rectangle in the first quadrant with sides parallel to the axes. If the width of $T$ is an integer, prove that $T$ has equal areas of black and white. Note that a similar argument works to show that if the height of $T$ is an integer, then $T$ has equal areas of black and white.
(b) Let $R$ be a rectangle with vertices at $(0, 0)$, $(a, 0)$, $(a, b)$, and $(0, b)$ with $a$ and $b$ positive. If $R$ has equal areas of black and white, prove that either $a$ is an integer or that $b$ is an integer.
(c) Suppose a rectangle $R$ is tiled by a finite number of rectangular tiles. That is, the rectangular tiles completely cover $R$ but intersect only along their edges. If each of the tiles has at least one integer side, prove that $R$ has at least one integer side.
[b]p4.[/b] Call a number [i]simple [/i] if it can be expressed as a product of single-digit numbers (in base ten).
(a) Find two simple numbers whose sum is $2014$ or prove that no such numbers exist.
(b) Find a simple number whose last two digits are $37$ or prove that no such number exists.
[b]p5.[/b] Consider triangles for which the angles $\alpha$, $\beta$, and $\gamma$ form an arithmetic progression. Let $a, b, c$ denote the lengths of the sides opposite $\alpha$, $\beta$, $\gamma$ , respectively. Show that for all such triangles, $$\frac{a}{c}\sin 2\gamma +\frac{c}{a} \sin 2\alpha$$ has the same value, and determine an algebraic expression for this value.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1973
[b]p1.[/b] Solve the system of equations
$$xy = 2x + 3y$$
$$yz = 2y + 3z$$
$$zx =2z+3x$$
[b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers.
[b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$.
[b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor).
[b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$.
[img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1993
[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}$ and $\begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column.
a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point.
b. Write down a $2 \times 2$ matrix which has no saddle point
c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point.
[b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$.
b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$.
[b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$.
a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions.
b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$.
[b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$.
a. Decide, with proof, if there are infinitely many Pythagorean triples.
b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$.
c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$.
[b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$.
a. Find an example giving $s > 1$.
b. Prove that for any positive $x$ and $y,s <2$.
c. Find, with proof, the largest possible value of $s$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1984
[b]p1.[/b] For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer?
[b]p2.[/b] Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd.
[b]p3.[/b] Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$-axis such that the angle $XM$ makes with the negative end of the $x$-axis is twice the angle $YM$ makes with the positive end of the $x$-axis.
[b]p4.[/b] Let $a,b$ be positive integers such that $a \ge b \sqrt3$. Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$.
i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists.
ii. Evaluate this limit.
[b]p5.[/b] Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$, so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$. Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles.
Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1968
[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)?
[b]p2.[/b] Prove that any triangle can be cut into three pieces which can be rearranged to form a rectangle of the same area.
[b]p3.[/b] An increasing sequence of integers starting with $1$ has the property that if $n$ is any member of the sequence, then so also are $3n$ and $n + 7$. Also, all the members of the sequence are solely generated from the first nummber $1$; thus the sequence starts with $1,3,8,9,10, ...$ and $2,4,5,6,7...$ are not members of this sequence. Determine all the other positive integers which are not members of the sequence.
[b]p4.[/b] Three prime numbers, each greater than $3$, are in arithmetic progression. Show that their common difference is a multiple of $6$.
[b]p5.[/b] Prove that if $S$ is a set of at least $7$ distinct points, no four in a plane, the volumes of all the tetrahedra with vertices in $S$ are not all equal.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2011
[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$.
[img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img]
[b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$.
(a) Give an example of a lucky set with five elements.
(b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote]
[b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that
(a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$,
(b) $p(x) > 0$ for all real $x$,
(c) $q(x) > 0$ for all real $x$.
[b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions.
[b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].