Found problems: 55
MMPC Part II 1996 - 2019, 2006
[b]p1.[/b] Suppose $A$, $B$ and $C$ are the angles of a triangle. Prove that
$$1 - 8 \cos A\cos B \cos C = sin^2(B - C) + (cos(B - C) - 2 cosA)^2.$$
[b]p2.[/b] Let $x_1, x_2,..., x_{100}$ be integers whose values are either $0$ or $1$.
(a) Show that $$x_1 + x_2 + ... + x_{100} - (x_1x_2 + x_2x_3 + ... + x_{99}x_{100} + x_{100}x_1)\le 50.$$
(b) Give specific values for $x_1, x_2,..., x_{100}$ that give equality.
[b]p3.[/b] Let $ABCD$ be a trapezoid whose area is $32$ square meters. Suppose the lengths of the parallel segments $AB$ and $DC$ are $2$ meters and $6$ meters, respectively, and $P$ is the intersection of the diagonals $AC$ and $BD$. If a line through $P$ intersects $AD$ and $BC$ at $E$ and $F$, respectively, determine, with a proof, the minimum possible area for quadrilateral $ABFE$.
[b]p4.[/b] Let $n$ be a positive integer and $x$ be a real number. Show that
$$\lfloor nx \rfloor = \lfloor x \rfloor +\left\lfloor x + \frac{1}{n} \right\rfloor + \left\lfloor x + \frac{2}{n} \right\rfloor + ... + \left\lfloor x + \frac{n - 1}{n} \right\rfloor$$
where $\lfloor a \rfloor$ is the greatest integer less than or equal to $a$. (For example, $\lfloor 4.5\rfloor = 4$ and $\lfloor - 4.5 \rfloor = -5$.)
[b]p5.[/b] A $3n$-digit positive integer (in base $10$) containing no zero is said to be [i]quad-perfect[/i] if the number is a perfect square and each of the three numbers obtained by viewing the first $n$ digits, the middle $n$ digits and the last $n$ digits as three $n$-digit numbers is in itself a perfect square. (For example, when $n = 1$, the only quad-perfect numbers are $144$ and $441$.) Find all $9$-digit quad-perfect numbers.
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MMPC Part II 1996 - 2019, 1998
[b]p1.[/b] An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left(\frac{p}{4}\right)$ for $p > 61$ plates.
a) For what number of plates $N$ does it become at least as cheap to use the second caterer as the first?
b) Let $N$ be the number you found in a). For what number of plates $X$ is the second caterer's price exactly double the price for $N$ plates?
c) Let $X$ be the number you found in b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer?
[b]p2.[/b] Let $N$ be a positive integer. Prove the following:
a) If $N$ is divisible by $4$, then $N$ can be expressed as the sum of two or more consecutive odd integers.
b) If $N$ is a prime number, then $N$ cannot be expressed as the sum of two or more consecutive odd integers.
c) If $N$ is twice some odd integer, then $N$ cannot be expressed as the sum of two or more consecutive odd integers.
[b]p3.[/b] Let $S =\frac{1}{1^2} +\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$
a) Find, in terms of $S$, the value of $S =\frac{1}{2^2} +\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...$
b) Find, in terms of $S$, the value of$S =\frac{1}{1^2} +\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...$
c) Find, in terms of $S$, the value of$S =\frac{1}{1^2} -\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...$
[b]p4.[/b] Let $\{P_1, P_2, P_3, ...\}$ be an infinite set of points on the $x$-axis having positive integer coordinates, and let $Q$ be an arbitrary point in the plane not on the $x$-axis. Prove that infinitely many of the distances $|P_iQ|$ are not integers.
a) Draw a relevant picture.
b) Provide a proof.
[b]p5.[/b] Point $P$ is an arbitrary point inside triangle $ABC$. Points $X$, $Y$ , and $Z$ are constructed to make segments $PX$, $PY$ , and $PZ$ perpendicular to $AB$, $BC$, and $CA$, respectively. Let $x$, $y$, and $z$ denote the lengths of the segments $PX$, $PY$ , and $PZ$, respectively.
a) If triangle $ABC$ is an equilateral triangle, prove that $x + y + z$ does not change regardless of the location of $P$ inside triangle ABC.
b) If triangle $ABC$ is an isosceles triangle with $|BC| = |CA|$, prove that $x + y + z$ does not change when $P$ moves along a line parallel to $AB$.
c) Now suppose that triangle $ABC$ is scalene (i.e., $|AB|$, $|BC|$, and $|CA|$ are all different). Prove that there exists a line for which $x+y+z$ does not change when $P$ moves along this line.
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MMPC Part II 1958 - 95, 1994
[b]p1.[/b] Al usually arrives at the train station on the commuter train at $6:00$, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane arrived home $12$ minutes earlier than usual. If Al was jogging at a constant speed of $5$ miles per hour, and Jane always drives at the constant speed that would put her at the station at $6:00$, what was her speed, in miles per hour?
[b]p2.[/b] In the figure, points $M$ and $N$ are the respective midpoints of the sides $AB$ and $CD$ of quadrilateral $ABCD$. Diagonal $AC$ meets segment $MN$ at $P$, which is the midpoint of $MN$, and $AP$ is twice as long as $PC$. The area of triangle $ABC$ is $6$ square feet.
(a) Find, with proof, the area of triangle $AMP$.
(b) Find, with proof, the area of triangle $CNP$.
(c) Find, with proof, the area of quadrilateral $ABCD$.
[img]https://cdn.artofproblemsolving.com/attachments/a/c/4bdcd8390bae26bc90fc7eae398ace06900a67.png[/img]
[b]p3.[/b] (a) Show that there is a triangle whose angles have measure $\tan^{-1}1$, $\tan^{-1}2$ and $\tan^{-1}3$.
(b) Find all values of $k$ for which there is a triangle whose angles have measure $\tan^{-1}\left(\frac12 \right)$, $\tan^{-1}\left(\frac12 +k\right)$, and $\tan^{-1}\left(\frac12 +2k\right)$
[b]p4.[/b] (a) Find $19$ consecutive integers whose sum is as close to $1000$ as possible.
(b) Find the longest possible sequence of consecutive odd integers whose sum is exactly $1000$, and prove that your sequence is the longest.
[b]p5.[/b] Let $AB$ and $CD$ be chords of a circle which meet at a point $X$ inside the circle.
(a) Suppose that $\frac{AX}{BX}=\frac{CX}{DX}$. Prove that $|AB|=|CD|$.
(b) Suppose that $\frac{AX}{BX}>\frac{CX}{DX}>1$. Prove that $|AB|>|CD|$.
($|PQ|$ means the length of the segment $PQ$.)
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MMPC Part II 1996 - 2019, 2015
[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last August, Jennifer McLoud-Mann, along with her husband Casey Mann and an undergraduate David Von Derau at the University of Washington, Bothell, discovered a new tiling pattern of the plane with a pentagon. This is the fifteenth pattern of using a pentagon to cover the plane with no gaps or overlaps. It is unknown whether other pentagons tile the plane, or even if the number of patterns is finite. Below is a portion of this new tiling pattern.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS8xLzM4M2RjZDEzZTliYTlhYTJkZDU4YTA4ZGMwMTA0MzA5ODk1NjI0LnBuZw==&rn=bW1wYyAyMDE1LnBuZw==[/img]
Determine the five angles (in degrees) of the pentagon $ABCDE$ used in this tiling. Explain your reasoning, and give the values you determine for the angles at the bottom.
[b]p3.[/b] Let $f(x) =\sqrt{2019 + 4\sqrt{2015}} +\sqrt{2015} x$. Find all rational numbers $x$ such that $f(x)$ is a rational number.
[b]p4.[/b] Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process.
$\bullet$ Let the numbers on the whiteboard be $a$ and $b$, with $a \le b$.
$\bullet$ Write $a^2$ on the blackboard.
$\bullet$ Erase $b$ from the whiteboard and replace it with $b - a$.
For example, if the whiteboard began with 5 and 8, Alice first writes $25$ on the blackboard and changes the whiteboard to $5$ and $3$. Her next move is to write $9$ on the blackboard and change the whiteboard to $2$ and $3$.
Alice stops when one of the numbers on the whiteboard is 0. At this point the sum of the numbers on the blackboard is $2015$.
a. If one of the starting numbers is $1$, what is the other?
b. What are all possible starting pairs of numbers?
[b]p5.[/b] Professor Beatrix Quirky has many multi-volume sets of books on her shelves. When she places a numbered set of $n$ books on her shelves, she doesn’t necessarily place them in order with book $1$ on the left and book $n$ on the right. Any volume can be placed at the far left. The only rule is that, except the leftmost volume, each volume must have a volume somewhere to its left numbered either one more or one less. For example, with a series of six volumes, Professor Quirky could place them in the order $123456$, or $324561$, or $564321$, but not $321564$ (because neither $4$ nor $6$ is to the left of $5$).
Let’s call a sequence of numbers a [i]quirky [/i] sequence of length $n$ if:
1. the sequence contains each of the numbers from $1$ to $n$, once each, and
2. if $k$ is not the first term of the sequence, then either $k + 1$ or $k - 1$ occurs somewhere before $k$ in the sequence.
Let $q_n$ be the number of quirky sequences of length $n$. For example, $q_3 = 4$ since the quirky sequences of length $3$ are $123$, $213$, $231$, and $321$.
a. List all quirky sequences of length $4$.
b. Find an explicit formula for $q_n$. Prove that your formula is correct.
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MMPC Part II 1958 - 95, 1988
[b]p1.[/b] Given an equilateral triangle $ABC$ with area $16\sqrt3$, and an interior point $P$ with distances from vertices $|AP| = 4$ and $|BP| = 6$.
(a) Find the length of each side.
(b) Find the distance from point $P$ to the side $AB$.
(c) Find the distance $|PC|$.
[b]p2.[/b] Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them $A$ (who tosses first), $B$ (second), $C$ (third, if there is a third), etc.
(a) If there are only two players, what is the probability that $A$ (the first) wins?
(b) If there are exactly $3$ players, what is the probability that $A$ (the first) wins?
(c) If there are exactly $3$ players, what is the probability that $B$ (the second) wins?
[b]p3.[/b] A circular castle of radius $r$ is surrounded by a circular moat of width $m$ ($m$ is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat.
(a) If the radius $r$ is $140$ feet and there are only $3$ life guards available, what is the minimum possible width of moat they can watch?
(b) Find the minimum number of life guards needed as a function of $r$ and $m$.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png[/img]
[b]p4.[/b] (a)Find all linear (first degree or less) polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all linear polynomials $g(x)$.
(b) Prove your answer to part (a).
(c) Find all polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all polynomials $g(x)$.
(d) Prove your answer to part (c).
[b]p5.[/b] A non-empty set $B$ of integers has the following two properties:
i. each number $x$ in the set can be written as a sum $x = y+ z$ for some $y$ and $z$ in the set $B$. (Warning: $y$ and $z$ may or may not be distinct for a given $x$.)
ii. the number $0$ can not be written as a sum $0 = y + z$ for any $y$ and $z$ in the set $B$.
(a) Find such a set $B$ with exactly $6$ elements.
(b) Find such a set $B$ with exactly $6$ elements, and such that the sum of all the $6$ elements is $1988$.
(c) What is the smallest possible size of such a set $B$ ?
(d) Prove your answer to part (c).
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MMPC Part II 1958 - 95, 1964
[b]p1.[/b] The edges of a tetrahedron are all tangent to a sphere. Prove that the sum of the lengths of any pair of opposite edges equals the sum of the lengths of any other pair of opposite edges. (Two edges of a tetrahedron are said to be opposite if they do not have a vertex in common.)
[b]p2.[/b] Find the simplest formula possible for the product of the following $2n - 2$ factors: $$\left(1+\frac12 \right),\left(1-\frac12 \right), \left(1+\frac13 \right) , \left(1-\frac13 \right),...,\left(1+\frac{1}{n} \right), \left(1-\frac{1}{n} \right)$$. Prove that your formula is correct.
[b]p3.[/b] Solve $$\frac{(x + 1)^2+1}{x + 1} + \frac{(x + 4)^2+4}{x + 4}=\frac{(x + 2)^2+2}{x + 2}+\frac{(x + 3)^2+3}{x + 3}$$
[b]p4.[/b] Triangle $ABC$ is inscribed in a circle, $BD$ is tangent to this circle and $CD$ is perpendicular to $BD$. $BH$ is the altitude from $B$ to $AC$. Prove that the line $DH$ is parallel to $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/e/9/4d0b136dca4a9b68104f00300951837adef84c.png[/img]
[b]p5.[/b] Consider the picture below as a section of a city street map. There are several paths from $A$ to $B$, and if one always walks along the street, the shortest paths are $15$ blocks in length. Find the number of paths of this length between $A$ and $B$.
[img]https://cdn.artofproblemsolving.com/attachments/8/d/60c426ea71db98775399cfa5ea80e94d2ea9d2.png[/img]
[b]p6.[/b] A [u]finite [/u] [u]graph [/u] is a set of points, called [u]vertices[/u], together with a set of arcs, called [u]edges[/u]. Each edge connects two of the vertices (it is not necessary that every pair of vertices be connected by an edge). The [u]order [/u] of a vertex in a finite graph is the number of edges attached to that vertex.
[u]Example[/u]
The figure at the right is a finite graph with $4$ vertices and $7$ edges. [img]https://cdn.artofproblemsolving.com/attachments/5/9/84d479c5dbd0a6f61a66970e46ab15830d8fba.png[/img]
One vertex has order $5$ and the other vertices order $3$.
Define a finite graph to be [u]heterogeneous [/u] if no two vertices have the same order.
Prove that no graph with two or more vertices is heterogeneous.
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MMPC Part II 1996 - 2019, 1996
[b]p1.[/b] An Egyptian fraction has the form $1/n$, where $n$ is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, $3/5$ was seen as $1/2 + 1/10$, or $1/3 + 1/4 + 1/60$. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for $3/5$ would be $1/2 + 1/10$.
a) Find the greedy Egyptian fraction representations for $2/13$.
b) Find the greedy Egyptian fraction representations for $9/10$.
c) Find the greedy Egyptian fraction representations for $2/(2k+1)$, where $k$ is a positive integer.
d) Find the greedy Egyptian fraction representations for $3/(6k+1)$, where $k$ is a positive integer.
[b]p2.[/b] a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii.
[img]https://cdn.artofproblemsolving.com/attachments/8/1/7c4d81ebfbd4445dc31fa038d9dc68baddb424.png[/img]
b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number $d$. The area of the larger triangle is twice the area of the smaller triangle. Find $d$.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/1f0d56d8e9e42574053c831fa129eb40c093d9.png[/img]
[b]p3.[/b] Suppose that the domain of a function $f$ is the set of real numbers and that $f$ takes values in the set of real numbers. A real number $x_0$ is a fixed point of f if $f(x_0) = x_0$.
a) Let $f(x) = m x + b$. For which $m$ does $f$ have a fixed point?
b) Find the fixed point of f$(x) = m x + b$ in terms of m and b, when it exists.
c) Consider the functions $f_c(x) = x^2 - c$.
i. For which values of $c$ are there two different fixed points?
ii. For which values of $c$ are there no fixed points?
iii. In terms of $c$, find the value(s) of the fixed point(s).
d) Find an example of a function that has exactly three fixed points.
[b]p4.[/b] A square based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level.
a) How many balls are there in the pyramid?
b) If each ball has a radius of $1$ meter, how tall is the pyramid?
c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls?
[b]p5.[/b] We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly $0$) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of $4$ suits, $13$ denominations, and $0$ jokers.
a) For a deck with $3$ suits $\{a, b, c\}$ and $7$ denominations $\{1, 2, 3, 4, 5, 6, 7\}$, and $0$ jokers, find the probability that a 3-card hand will be a straight. (A straight consists of $3$ cards in sequence, e.g., $1 \heartsuit$ ,$2 \spadesuit$ , $3\clubsuit$ , $2\diamondsuit$ but not $6 \heartsuit$ ,$7 \spadesuit$ , $1\diamondsuit$).
b) For a deck with $3$ suits, $7$ denominations, and $0$ jokers, find the probability that a $3$-card hand will consist of $3$ cards of the same suit (i.e., a flush).
c) For a deck with $3$ suits, $7$ denominations, and $1$ joker, find the probability that a $3$-card hand dealt at random will be a straight and also the probability that a $3$-card hand will be a flush.
d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain $1$ joker and is to have identical probabilities for a straight and a flush when a $3$-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur.
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MMPC Part II 1996 - 2019, 2007
[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$
where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$.
[b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions:
$\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions.
$\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions.
Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)?
[b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals.
[b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that
$$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$
for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$.
[b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$.
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MMPC Part II 1996 - 2019, 2005
[b]p1.[/b] Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle.
[b]p2.[/b] Determine the greatest integer that will divide $13,511$, $13,903$ and $14,589$ and leave the same remainder.
[b]p3.[/b] Suppose $A, B$ and $C$ are the angles of the triangle. Show that $\cos^2 A + \cos^2 B + \cos^2 C + 2 \cos A \cos B \cos C = 1$
[b]p4.[/b] Given the linear fractional transformation $f_1(x) =\frac{2x - 1}{x + 1}$.
Define $f_{n+1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3,...$ .
It can be shown that $f_{35} = f_5$.
(a) Find a function $g$ such that $f_1(g(x)) = g(f_1(x)) = x$.
(b) Find $f_{28}$.
[b]p5.[/b] Suppose $a$ is a complex number such that $a^{10} + a^5 + 1 = 0$. Determine the value of $a^{2005} + \frac{1}{a^{2005}}$.
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MMPC Part II 1958 - 95, 1979
[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$
[b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$.
[b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor.
[b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it.
Prove that the number of these smaller triangular regions is always odd.
[b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$.
[img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img]
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MMPC Part II 1958 - 95, 1961
[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$
$$2x + 4y + 17z = 301.$$
[b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$.
[b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$.
[b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$
$a, b, c, d, e, f$ real.
[b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1971
[b]p1[/b]. Prove that there is no interger $n$ such that $n^2 +1$ is divisible by $7$.
[b]p2.[/b] Find all solutions of the system
$$x^2-yz=1$$
$$y^2-xz=2$$
$$z^2-xy=3$$
[b]p3.[/b] A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number, $n$ , of points in the plane with the property that every three of them form the vertices of a triangle with long legs?
Prove all assertions.
[b]p4.[/b] Prove that the area of a quadrilateral of sides $a, b, c, d$ which can be inscribed in a circle and circumscribed about another circle is given by $A=\sqrt{abcd}$
[b]p5.[/b] Prove that all of the squares of side length $$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...$$ can fit inside a square of side length $1$ without overlap.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1958
[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$.
[b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals.
[b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum.
[b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$.
[b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1985
[b]p1.[/b] Sometimes one finds in an old park a tetrahedral pile of cannon balls, that is, a pile each layer of which is a tightly packed triangular layer of balls.
A. How many cannon balls are in a tetrahedral pile of cannon balls of $N$ layers?
B. How high is a tetrahedral pile of cannon balls of $N$ layers? (Assume each cannon ball is a sphere of radius $R$.)
[b]p2.[/b] A prime is an integer greater than $1$ whose only positive integer divisors are itself and $1$.
A. Find a triple of primes $(p, q, r)$ such that $p = q + 2$ and $q = r + 2$ .
B. Prove that there is only one triple $(p, q, r)$ of primes such that $p = q + 2$ and $q = r + 2$ .
[b]p3.[/b] The function $g$ is defined recursively on the positive integers by $g(1) =1$, and for $n>1$ , $g(n)= 1+g(n-g(n-1))$ .
A. Find $g(1)$ , $g(2)$ , $g(3)$ and $g(4)$ .
B. Describe the pattern formed by the entire sequence $g(1) , g(2 ), g(3), ...$
C. Prove your answer to Part B.
[b]p4.[/b] Let $x$ , $y$ and $z$ be real numbers such that $x + y + z = 1$ and $xyz = 3$ .
A. Prove that none of $x$ , $y$ , nor $z$ can equal $1$.
B. Determine all values of $x$ that can occur in a simultaneous solution to these two equations (where $x , y , z$ are real numbers).
[b]p5.[/b] A round robin tournament was played among thirteen teams. Each team played every other team exactly once. At the conclusion of the tournament, it happened that each team had won six games and lost six games.
A. How many games were played in this tournament?
B. Define a [i]circular triangle[/i] in a round robin tournament to be a set of three different teams in which none of the three teams beat both of the other two teams. How many circular triangles are there in this tournament?
C. Prove your answer to Part B.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2012
[b]p1.[/b] A permutation on $\{1, 2,..., n\}$ is an ordered arrangement of the numbers. For example, $32154$ is a permutation of $\{1, 2, 3, 4, 5\}$. Does there exist a permutation $a_1a_2... a_n$ of $\{1, 2,..., n\}$ such that $i+a_i$ is a perfect square for every $1 \le i \le n$ when
a) $n = 6$ ?
b) $n = 13$ ?
c) $n = 86$ ?
Justify your answers.
[b]p2.[/b] Circle $C$ and circle $D$ are tangent at point $P$. Line $L$ is tangent to $C$ at point $Q$ and to $D$ at point $R$ where $Q$ and $R$ are distinct from $P$. Circle $E$ is tangent to $C, D$, and $L$, and lies inside triangle $PQR$. $C$ and $D$ both have radius $8$. Find the radius of $E$, and justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/f/b/4b98367ea64e965369345247fead3456d3d18a.png[/img]
[b]p3.[/b] (a) Prove that $\sin 3x = 4 \cos^2 x \sin x - \sin x$ for all real $x$.
(b) Prove that $$(4 \cos^2 9^o - 1)(4 \cos^2 27^o - 1)(4 cos^2 81^o - 1)(4 cos^2 243^o - 1)$$ is an integer.
[b]p4.[/b] Consider a $3\times 3\times 3$ stack of small cubes making up a large cube (as with the small cubes in a Rubik's cube). An ant crawls on the surface of the large cube to go from one corner of the large cube to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal?
[b]p5.[/b] Let $m$ and $n$ be positive integers, and consider the rectangular array of points $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$. For what pairs m; n of positive integers does there exist a polygon for which the $mn$ points $(i, j)$ are its vertices, such that each edge is either horizontal or vertical? The figure below depicts such a polygon with $m = 10$, $n = 22$. Thus $10$, $22$ is one such pair.
[img]https://cdn.artofproblemsolving.com/attachments/4/5/c76c0fe197a8d1ebef543df8e39114fe9d2078.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1974
[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer.
[b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.)
[b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality:
(a) $xy \le \left(\frac{x+y}{2}\right)^2$
(b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$
(c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$
[b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common.
[img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img]
[b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2002
[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$.
(b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$.
[b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$.
(b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$.
[b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$.
(a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$.
(b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$.
[b]p4.[/b] In the Gregorian calendar:
(i) years not divisible by $4$ are common years,
(ii) years divisible by $4$ but not by $100$ are leap years,
(iii) years divisible by $100$ but not by $400$ are common years,
(iv) years divisible by $400$ are leap years,
(v) a leap year contains $366$ days, a common year $365$ days.
From the information above:
(a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks.
(b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$.
[b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$
The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1983
[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims.
[b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns.
Note that it is not required that he return to his starting point.
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img]
[b]p3.[/b] $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track?
[b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$.
Suggestion: Reflect!!
[b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers.
Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 1999
[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were
1. Michigan
2. Minnesota
З. Iowa
4. Indiana
5. Michigan State
6. Purdue
7. Northwestern
8. Ohio State
9. Penn State
10. Wisconsin
(Illinois does not participate in Women's Softball.)
When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places):
(Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin)
(Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State)
(Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin)
(Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern)
(Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State)
Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season.
If you cannot determine the standings, explain why you do not have enough information. You must justify your answer.
[b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$.
b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$.
c) Find the maximum value of $3 \sin t +2 \cos t$.
[b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes.
[b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible.
b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible.
[b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys.
a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room.
b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1969
[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$?
[b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots.
[b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common.
[b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete.
[b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1976
[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy
a) $3$ footballs and $2$ tennis balls?
b) $2$ footballs and $3$ tennis balls?
[b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle).
a) Prove that $AP \cdot PB = CP\cdot PD$.
b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle.
[b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$.
[b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one.
a) Find a formula expressing $x_n$ as a function of$ n$.
b) Prove your result.
[b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$.
a) Prove that $AM \le \frac12 AB + \frac12 AC$.
b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1986
[b]p1.[/b] $\vartriangle DEF$ is constructed from equilateral $\vartriangle ABC$ by choosing $D$ on $AB$, $E$ on $BC$ and $F$ on $CA$ so that $\frac{DB}{AB}=\frac{EC}{BC}=\frac{FA}{CA}=a$, where $a$ is a number between $0$ and $1/2$.
(a) Show that $\vartriangle DEF$ is also equilateral.
(b) Determine the value of $a$ that makes the area of $\vartriangle DEF$ equal to one half the area of $\vartriangle ABC$.
[b]p2.[/b] A bowl contains some red balls and some white balls. The following operation is repeated until only one ball remains in the bowl:
Two balls are drawn at random from the bowl. If they have different colors, then the red one is discarded and the white one is returned to the bowl. If they have the same color, then both are discarded and a red ball (from an outside supply of red balls) is added to the bowl.
(Note that this operation—in either case—reduces the number of balls in the bowl by one.)
(a) Show that if the bowl originally contained exactly $1$ red ball and $ 2$ white balls, then the color of the ball remaining at the end (i.e., after two applications of the operation) does not depend on chance, and determine the color of this remaining ball.
(b) Suppose the bowl originally contained exactly $1986$ red balls and $1986$ white balls. Show again that the color of the ball remaining at the end does not depend on chance and determine its color.
[b]p3.[/b] Let $a, b$, and $c$ be three consecutive positive integers, with $a < b < c.$
(a) Show that $ab$ cannot be the square of an integer.
(b) Show that $ac$ cannot be the square of an integer.
(c) Show that $abc$ cannot be the square of an integer.
[b]p4.[/b] Consider the system of equations $$\sqrt{x}+\sqrt{y}=2$$
$$ x^2+y^2=5$$
(a) Show (algebraically or graphically) that there are two or more solutions in real numbers $x$ and $y$.
(b) The graphs of the two given equations intersect in exactly two points. Find the equation of the straight line passing through these two points of intersection.
[b]p5.[/b] Let $n$ and $m$ be positive integers. An $n \times m $ rectangle is tiled with unit squares. Let $r(n, m)$ denote the number of rectangles formed by the edges of these unit squares. Thus, for example, $r(2, 1) = 3$.
(a) Find $r(2, 3)$.
(b) Find $r(n, 1)$.
(c) Find, with justification, a formula for $r(n, m)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1972
[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles.
[img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img]
[b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by:
$D(n) = n$ if $1 \le n \le 9$
$D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$
where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ .
[b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ .
[b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians.
[img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img]
[b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1977
[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then?
[b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs.
[b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers:
$ 1 + 2 + 3- 4 + 5 = 7$
$-1 + 2 - 3- 4 + 5 = -1$
$1 + 2 + 3 + 4 + 5 = 15$ , etc.
[b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular.
[img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img]
[b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ .
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1996 - 2019, 2009
[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person.
(a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable.
(b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable.
[b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$?
[b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$.
(b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions.
[b]p4.[/b] Let function $f(x, y, z)$ be defined as following:
$$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$
Find the minimum value and prove the result.
[b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.