Found problems: 25
Maryland University HSMC part II, 2010
[b]p1.[/b] We say that six positive integers form a magic triangle if they are arranged in a triangular array as in the figure below in such a way that each number in the top two rows is equal to the sum of its two neighbors in the row directly below it. The triangle shown is magic because $4 = 1 + 3$, $5 = 3 + 2$, and $9 = 4 + 5$.
$$9$$
$$4\,\,\,\,5$$
$$1\,\,\,\,3\,\,\,\,2$$
(a) Find a magic triangle such that the numbers at the three corners are $10$, $20$, and $2010$, with $2010$ at the top.
(b) Find a magic triangle such that the numbers at the three corners are $20$, $201$, and $2010$, with $2010$ at the top, or prove that no such triangle exists.
[b]p2.[/b] (a) The equalities $\frac12+\frac13+\frac16= 1$ and $\frac12+\frac13+\frac17+\frac{1}{42}= 1$ express $1$ as a sum of the reciprocals
of three (respectively four) distinct positive integers. Find five positive integers $a < b < c <d < e$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1.$$
(b) Prove that for any integer $m \ge 3$, there exist $m$ positive integers $d_1 < d_2 <... < d_m$ such that $$\frac{1}{d_1}+\frac{1}{d_2}+ ... +\frac{1}{d_m}= 1.$$
[b]p3.[/b] Suppose that $P(x) = a_nx^n +... + a_1x + a_0$ is a polynomial of degree n with real coefficients. Say that the real number $b$ is a balance point of $P$ if for every pair of real numbers $a$ and $c$ such that $b$ is the average of $a$ and $c$, we have that $P(b)$ is the average of $P(a)$ and $P(c)$. Assume that $P$ has two distinct balance points. Prove that $n$ is at most $1$, i.e., that $P$ is a linear function.
[b]p4.[/b] A roller coaster at an amusement park has a train consisting of $30$ cars, each seating two people next to each other. $60$ math students want to take as many rides as they can, but are told that there are two rules that cannot be broken. First, all $60$ students must ride each time, and second, no two students are ever allowed to sit next to each other more than once. What is the maximal number of roller coaster rides that these students can take? Justify your answer.
[b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that the lengths of all four sides and the two diagonals of $ABCD$ are rational numbers. If the two diagonals $AC$ and $BD$ intersect at a point $M$, prove that the length of $AM$ is also a rational number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2004
[b]p1.[/b] Archimedes, Euclid, Fermat, and Gauss had a math competition.
Archimedes said, “I did not finish $1$st or $4$th.”
Euclid said, “I did not finish $4$th.”
Fermat said, “I finished 1st.” Gauss said, “I finished $4$th.”
There were no ties in the competition, and exactly three of the mathematicians told the truth.
Who finished first and who finished last? Justify your answers.
[b]p2.[/b] Find the area of the set in the xy-plane defined by $x^2 - 2|x| + y^2 \le 0$. Justify your answer.
[b]p3.[/b] There is a collection of $2004$ circular discs (not necessarily of the same radius) in the plane. The total area covered by the discs is $1$ square meter. Show that there is a subcollection $S$ of discs such that the discs in S are non-overlapping and the total area of the discs in $S$ is at least $1/9$ square meter.
[b]p4.[/b] Let $S$ be the set of all $2004$-digit integers (in base $10$) all of whose digits lie in the set $\{1, 2, 3, 4\}$. (For example, $12341234...1234$ is in $S$.) Let $n_0$ be the number of $s \in S$ such that $s$ is a multiple of $3$, let $n_1$ be the number of $s \in S$ such that $s$ is one more than a multiple of $3$, and let $n_2$ be the number of $s \in S$ such that $s$ is two more than a multiple of $3$. Determine which of $n_0$, $n_1$, $n_2$ is largest and which is smallest (and if there are any equalities). Justify your answers.
[b]p5.[/b] There are $6$ members on the Math Competition Committee. The problems are kept in a safe. There are $\ell$ locks on the safe and there are $k$ keys, several for each lock. The safe does not open unless all of the locks are unlocked, and each key works on exactly one lock. The keys should be distributed to the $6$ members of the committee so that each group of $4$ members has enough keys to open all of the $\ell$ locks. However, no group of $3$ members should be able to open all of the $\ell$ locks.
(a) Show that this is possible with $\ell = 20$ locks and $k = 60$ keys. That is, it is possible to use $20$ locks and to choose and distribute 60 keys in such a way that every group of $4$ can open the safe, but no group of $3$ can open the safe.
(b) Show that we always must have $\ell \ge 20$ and $k\ge60$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 1999
[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has either $19$ or $21$ cupcakes).
a) Find a way to do this.
b) Suppose a Y2K bug eats one of the cupcakes, so you have only $1999$ cupcakes. Show that it is impossible to arrange the cupcakes on the tables according to the above conditions.
[b]p2.[/b] Let $P$ and $Q$ lie on the hypotenuse $AB$ of the right triangle $CAB$ so that $|AP|=|PQ|=|QB|=|AB|/3$. Suppose that $|CP|^2+|CQ|^2=5$. Prove that $|AB|$ has the same value for all such triangles, and find that value. Note: $|XY|$ denotes the length of the segment $XY$.
[b]p3.[/b] Let $P$ be a polynomial with integer coefficients and let $a, b, c$ be integers. Suppose $P(a)=b$, $P(b)=c$, and $P(c)=a$. Prove that $a=b=c$.
[b]p4.[/b] A lattice point is a point $(x,y)$ in the plane for which both $x$ and $y$ are integers. Each lattice point is painted with one of $1999$ available colors. Prove that there is a rectangle (of nonzero height and width) whose corners are lattice points of the same color.
[b]p5.[/b] A $1999$-by-$1999$ chocolate bar has vertical and horizontal grooves which divide it into $1999^2$ one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then either putting both rectangles down or eating one piece and putting the other piece down. The players move alternately. The one who cannot make a move at his turn (because there are only one-by-one squares left) loses. Caesar starts. Which player has a winning strategy? Describe a winning strategy for that player.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 1998
[b]p1.[/b] Four positive numbers are placed at the vertices of a rectangle. Each number is at least as large as the average of the two numbers at the adjacent vertices. Prove that all four numbers are equal.
[b]p2.[/b] The sum $498+499+500+501=1998$ is one way of expressing $1998$ as a sum of consecutive positive integers. Find all ways of expressing $1998$ as a sum of two or more consecutive positive integers. Prove your list is complete.
[b]p3.[/b] An infinite strip (two parallel lines and the region between them) has a width of $1$ inch. What is the largest value of $A$ such that every triangle with area $A$ square inches can be placed on this strip? Justify your answer.
[b]p4.[/b] A plane divides space into two regions. Two planes that intersect in a line divide space into four regions. Now suppose that twelve planes are given in space so that
a) every two of them intersect in a line,
b) every three of them intersect in a point, and
c) no four of them have a common point.
Into how many regions is space divided? Justify your answer.
[b]p5.[/b] Five robbers have stolen $1998$ identical gold coins. They agree to the following: The youngest robber proposes a division of the loot. All robbers, including the proposer, vote on the proposal. If at least half the robbers vote yes, then that proposal is accepted. If not, the proposer is sent away with no loot and the next youngest robber makes a new proposal to be voted on by the four remaining robbers, with the same rules as above. This continues until a proposed division is accepted by at least half the remaining robbers. Each robber guards his best interests: He will vote for a proposal if and only if it will give him more coins than he will acquire by rejecting it, and the proposer will keep as many coins for himself as he can. How will the coins be distributed? Explain your reasoning.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2017
[b]p1[/b]. Consider the following four statements referring to themselves:
1. At least one of these statements is true.
2. At least two of these statements are false.
3. At least three of these statements are true.
4. All four of these statements are false.
Determine which statements are true and which are false. Justify your answer.
[b]p2.[/b] Let $f(x) = a_{2017}x^{2017} + a_{2016}x^{2016} + ... + a_1x + a_0$ where the coefficients $a_0, a_1, ... , a_{2017}$ have not yet been determined. Alice and Bob play the following game:
$\bullet$ Alice and Bob alternate choosing nonzero integer values for the coefficients, with Alice going first. (For example, Alice’s first move could be to set $a_{18}$ to $-3$.)
$\bullet$ After all of the coefficients have been chosen:
- If f(x) has an integer root then Alice wins.
- If f(x) does not have an integer root then Bob wins.
Determine which player has a winning strategy and what the strategy is. Make sure to justify your answer.
[b]p3.[/b] Suppose that a circle can be inscribed in a polygon $P$ with $2017$ equal sides. Prove that $P$ is a regular polygon; that is, all angles of $P$ are also equal.
[b]p4.[/b] A $3 \times 3 \times 3$ cube of cheese is sliced into twenty-seven $ 1 \times 1 \times 1$ blocks. A mouse starts anywhere on the outside and eats one of the $1\times 1\times 1$ cubes. He then moves to an adjacent cube (in any direction), eats that cube, and continues until he has eaten all $27$ cubes. (Two cubes are considered adjacent if they share a face.) Prove that no matter what strategy the mouse uses, he cannot eat the middle cube last.
[Note: One should neglect gravity – intermediate configurations don’t collapse.]
p5. Suppose that a constant $c > 0$ and an infinite sequence of real numbers $x_0, x_1, x_2, ...$ satisfy
$x_{k+1} =\frac{x_k + 1}{1 - cx_k}$ for all $k \ge 0$. Prove that the sequence $x_0, x_1, x_2, ....$ contains infinitely many positive terms and also contains infinitely many negative terms.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2001
[b]p1.[/b] A band of pirates unloaded some number of treasure chests from their ship. The number of pirates was between $60$ and $69$ (inclusive). Each pirate handled exactly $11$ treasure chests, and each treasure chest was handled by exactly $7$ pirates. Exactly how many treasure chests were there? Show that your answer is the only solution.
[b]p2.[/b] Let $a$ and $b$ be the lengths of the legs of a right triangle, let $c$ be the length of the hypotenuse, and let $h$ be the length of the altitude drawn from the vertex of the right angle to the hypotenuse. Prove that $c+h>a+b$.
[b]p3.[/b] Prove that $$\frac{1}{70}< \frac{1}{2} \frac{3}{4} \frac{5}{6} ... \frac{2001}{2002} < \frac{1}{40}$$
[b]p4.[/b] Given a positive integer $a_1$ we form a sequence $a_1 , a_2 , a _3,...$ as follows: $a_2$ is obtained from $a_1$ by adding together the digits of $a_1$ raised to the $2001$-st power; $a_3$ is obtained from $a_2$ using the same rule, and so on.
For example, if $a_1 =25$, then $a_2 =2^{2001}+5^{2001}$, which is a $1399$-digit number containing $106$ $0$'s, $150$ $1$'s, 4124$ 42$'s, $157$ $3$'s, $148$ $4$'s, $141$ $5$'s, $128$ $6$'s, $1504 47$'s, $152$ $8$'s, $143$ $9$'s. So $a_3 = 106 \times 0^{2001}+ 150 \times 1^{2001}+ 124 \times 2^{2001}+ 157 \times 3^{2001}+ ...+ 143 \times 9^{2001}$ which is a $1912$-digit number, and so forth.
Prove that if any positive integer $a_1$ is chosen to start the sequence, then there is a positive integer $M$ (which depends on $a_1$ ) that is so large that $a_n < M$ for all $n=1,2,3,...$
[b]p5.[/b] Let $P(x)$ be a polynomial with integer coefficients. Suppose that there are integers $a$, $b$, and $c$ such that $P(a)=0$, $P(b)=1$, and $P(c)=2$. Prove that there is at most one integer $n$ such that $P(n)=4$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2003
[b]p1.[/b] (a) Find three positive integers $a, b, c$ whose sum is $407$, and whose product (when written in base $10$) ends in six $0$'s.
(b) Prove that there do NOT exist positive integers $a, b, c$ whose sum is $407$ and whose product ends in seven $0$'s.
[b]p2.[/b] Three circles, each of radius $r$, are placed on a plane so that the center of each circle lies on a point of intersection of the other two circles. The region $R$ consists of all points inside or on at least one of these three circles. Find the area of $R$.
[b]p3.[/b] Let $f_1(x) = a_1x^2+b_1x+c_1$, $f_2(x) = a_2x^2+b_2x+c_2$ and $f_3(x) = a_3x^2+b_3x+c_3$ be the equations of three parabolas such that $a_1 > a_2 > a-3$. Prove that if each pair of parabolas intersects in exactly one point, then all three parabolas intersect in a common point.
[b]p4.[/b] Gigafirm is a large corporation with many employees.
(a) Show that the number of employees with an odd number of acquaintances is even.
(b) Suppose that each employee with an even number of acquaintances sends a letter to each of these acquaintances. Each employee with an odd number of acquaintances sends a letter to each non-acquaintance. So far, Leslie has received $99$ letters. Prove that Leslie will receive at least one more letter.
(Notes: "acquaintance" and "non-acquaintance" refer to employees of Gigaform. If $A$ is acquainted with $B$, then $B$ is acquainted with $A$. However, no one is acquainted with himself.)
[b]p5.[/b] (a) Prove that for every positive integer $N$, if $A$ is a subset of the numbers $\{1, 2, ...,N\}$ and $A$ has size at least $2N/3 + 1$, then $A$ contains a three-term arithmetic progression (i.e., there are positive integers $a$ and $b$ so that all three of the numbers $a$,$a + b$, and $a + 2b$ are elements of $A$).
(b) Show that if $A$ is a subset of $\{1, 2, ..., 3500\}$ and $A$ has size at least $2003$, then $A$ contains a three-term arithmetic progression.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2019
[b]p1.[/b] Alex and Sam have a friend Pat, who is younger than they are. Alex, Sam and Pat all share a birthday. When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.)
[b]p2.[/b] Let $ABCD$ be a square with side length $2$. The four sides of $ABCD$ are diameters of four semicircles, each of which lies inside the square. The set of all points which lie on or inside two of these semicircles is a four petaled flower. Find (with proof) the area of this flower.
[img]https://cdn.artofproblemsolving.com/attachments/5/5/bc724b9f74c3470434c322020997a533986d33.png[/img]
[b]p3.[/b] A prime number is called [i]strongly prime[/i] if every integer obtained by permuting its digits is also prime. For example $113$ is strongly prime, since $113$, $131$, and $311$ are all prime numbers. Prove that there is no strongly prime number such that each of the digits $1, 3, 7$, and $9$ appears at least once in its decimal representation.
[b]p4.[/b] Suppose $n$ is a positive integer. Let an be the number of permutations of $1, 2, . . . , n$, where $i$ is not in the $i$-th position, for all $i$ with $1 \le i \le n$. For example $a_3 = 2$, where the two permutations that are counted are $231$, and $312$. Let bn be the number of permutations of $1, 2, . . . , n$, where no $i$ is followed by $i + 1$, for all $i$ with $1 \le i \le n - 1$. For example $b_3 = 3$, where the three permutations that are counted are $132$, $213$, and $321$. For every $n \ge 1$, find (with proof) a simple formula for $\frac{a_{n+1}}{b_n}$. Your formula should not involve summations. Use your formula to evaluate $\frac{a_{2020}}{b_{2019}}$.
[b]p5.[/b] Let $n \ge 2$ be an integer and $a_1, a_2, ... , a_n$ be positive real numbers such that $a_1 + a_2 +... + a_n = 1$. Prove that $$\sum^n_{k=1}\frac{a_k}{1 + a_{k+1} - a_{k-1}}\ge 1.$$
(Here $a_0 = a_n$ and $a_{n+1} = a_1$.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2018
[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you?
[b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle.
[b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties:
• $0 \in S$
• If $x \in S$ then $n - 2 - x \in S$
• If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$.
Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime.
(Note: “$\in$” means “belongs to”.)
[b]p4.[/b] Consider the $3 \times 3$ grid shown below
$\begin{tabular}{|l|l|l|l|}
\hline
A & B & C \\ \hline
D & E & F \\ \hline
G & H & I \\ \hline
\end{tabular}$
A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move.
Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.)
[b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2013
[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array.
[img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img]
[b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$.
[img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img]
[b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img]
[b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ .
[b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2007
[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles.
[b]p2.[/b] There was a young lady named Chris,
Who, when asked her age, answered this:
"Two thirds of its square
Is a cube, I declare."
Now what was the age of the miss?
(a) Find the smallest possible age for Chris. You must justify your answer.
(Note: ages are positive integers; "cube" means the cube of a positive integer.)
(b) Find the second smallest possible age for Chris. You must justify your answer.
(Ignore the word "young.")
[b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$
[b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$.
(b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram.
(Note: "distance to a side" means the shortest distance to the line obtained by extending the side.)
[b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2009
[b]p1.[/b] (a) Show that for every set of three integers, we can find two of them whose average is also an integer.
(b) Show that for every set of $5$ integers, there is a subset of three of them whose average is an integer.
[b]p2.[/b] Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two different quadratic polynomials such that $f(7) + f(11) = g(7) + g(11)$.
(a) Show that $f(9) = g(9)$.
(b) Show that $x = 9$ is the only value of $x$ where $f(x) = g(x)$.
[b]p3.[/b] Consider a rectangle $ABCD$ and points $E$ and $F$ on the sides $BC$ and $CD$, respectively, such that the areas of the triangles $ABE$, $ECF$, and $ADF$ are $11$, $3$, and $40$, respectively. Compute the area of rectangle $ABCD$.
[img]https://cdn.artofproblemsolving.com/attachments/f/0/2b0bb188a4157894b85deb32d73ab0077cd0b7.png[/img]
[b]p4.[/b] How many ways are there to put markers on a $8 \times 8$ checkerboard, with at most one marker per square, such that each of the $8$ rows and each of the $8$ columns contain an odd number of markers?
[b]p5.[/b] A robot places a red hat or a blue hat on each person in a room. Each person can see the colors of the hats of everyone in the room except for his own. Each person tries to guess the color of his hat. No communication is allowed between people and each person guesses at the same time (so no timing information can be used, for example). The only information a person has is the color of each other person’s hat.
Before receiving the hats, the people are allowed to get together and decide on their strategies. One way to think of this is that each of the $n$ people makes a list of all the possible combinations he could see (there are $2^{n-1}$ such combinations). Next to each combination, he writes what his guess will be for the color of his own hat. When the hats are placed, he looks for the combination on his list and makes the guess that is listed there.
(a) Prove that if there are exactly two people in the room, then there is a strategy that guarantees that always at least one person gets the right answer for his hat color.
(b) Prove that if you have a group of $2008$ people, then there is a strategy that guarantees that always at least $1004$ people will make a correct guess.
(c) Prove that if there are $2009$ people, then there is no strategy that guarantees that always at least $1005$ people will make a correct guess.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2022
[b]p1.[/b] Find a real number $x$ for which $x\lfloor x \rfloor = 1234.$
Note: $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
[b]p2.[/b] Let $C_1$ be a circle of radius $1$, and $C_2$ be a circle that lies completely inside or on the boundary of $C_1$. Suppose$ P$ is a point that lies inside or on $C_2$. Suppose $O_1$, and $O_2$ are the centers of $C_1$, and $C_2$, respectively. What is the maximum possible area of $\vartriangle O_1O_2P$? Prove your answer.
[b]p3.[/b] The numbers $1, 2, . . . , 99$ are written on a blackboard. We are allowed to erase any two distinct (but perhaps equal) numbers and replace them by their nonnegative difference. This operation is performed until a single number $k$ remains on the blackboard. What are all the possible values of $k$? Prove your answer.
Note: As an example if we start from $1, 2, 3, 4$ on the board, we can proceed by erasing $1$ and $2$ and replacing them by $1$. At that point we are left with $1, 3, 4$. We may then erase $3$ and $4$ and replacethem by $1$. The last step would be to erase $1$, $1$ and end up with a single $0$ on the board.
[b]p4.[/b] Let $a, b$ be two real numbers so that $a^3 - 6a^2 + 13a = 1$ and $b^3 - 6b^2 + 13b = 19$. Find $a + b$. Prove your answer.
[b]p5.[/b] Let $m, n, k$ be three positive integers with $n \ge k$. Suppose $A =\prod_{1\le i\le j\le m} gcd(n + i, k + j) $ is the product of $gcd(n + i, k + j)$, where $i, j$ range over all integers satisfying $1\le i\le j\le m$. Prove that the following fraction is an integer $$\frac{A}{(k + 1) \dots(k + m)}{n \choose k}.$$
Note: $gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ${n \choose k}= \frac{n!}{k!(n - k)!}$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2000
[b]p1.[/b] There are $2000$ cans of paint. Show that at least one of the following two statements must be true.
There are at least $45$ cans of the same color.
There are at least $45$ cans all of different colors.
[b]p2.[/b] The measures of the $3$ angles of one triangle are all different from each other but are the same as the measures of the $3$ angles of a second triangle. The lengths of $2$ sides of the first triangle are different from each other but are the same as the lengths of $2$ sides of the second triangle. Must the length of the remaining side of the first triangle be the same as the length of the remaining side of the second triangle? If yes, prove it. If not, provide an example.
[b]p3.[/b] Consider the sequence $a_1=1$, $a_2=2$, $a_3=5/2$, ... satisfying $a_{n+1}=a_n+(a_n)^{-1}$ for $n>1$. Show that $a_{10000}>141$.
[b]p4.[/b] Prove that no matter how $250$ points are placed in a disk of radius $1$, there is a disk of radius $1/10$ that contains at least $3$ of the points.
[b]p5.[/b] Prove that:
Given any $11$ integers (not necessarily distinct), one can select $6$ of them so that their sum is divisible by $6$.
Given any $71$ integers (not necessarily distinct), one can select $36$ of them so that their sum is divisible by $36$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2002
[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish.
[b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$.
[b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row.
[b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament.
[b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns.
a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty.
b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2012
[b]p1.[/b] (a) Suppose $101$ Dalmatians chase $2012$ squirrels. Each squirrel gets chased by at most one Dalmatian, and each Dalmatian chases at least one squirrel. Show that two Dalmatians chase the same number of squirrels.
(b) What is the largest number of Dalmatians that can chase $2012$ squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels?
[b]p2.[/b] Lucy and Linus play the following game. They start by putting the integers $1, 2, 3, ..., 2012$ in a hat. In each round of the game, Lucy and Linus each draw a number from the hat. If the two numbers are $a$ and $b$, they throw away these numbers and put the number $|a - b|$ back into the hat. After $2011$ rounds, there is only one number in the hat. If it is even, Lucy wins. If it is odd, Linus wins.
(a) Prove that there is a sequence of drawings that makes Lucy win.
(b) Prove that Lucy always wins.
[b]p3.[/b] Suppose $x$ is a positive real number and $x^{1990}$, $x^{2001}$, and $x^{2012}$ differ by integers. Prove that $x$ is an integer.
[b]p4.[/b] Suppose that each point in three-dimensional space is colored with one of five colors and suppose that each color is used at least once. Prove that there is some plane that contains at least four of the colors.
[b]p5.[/b] Two circles, $C_1$ and $C_2$, are tangent at point $A$, with $C_1$ lying inside $C_2$ (and $C_1 \ne C_2$). The line through their centers intersects $C_1$ at $B_1$ and $C_2$ at $B_2$. A line $L$ is drawn through $A$ and it intersects $C_1$ at $P_1$ (with $P_1 \ne A$) and intersects $C_2$ at $P_2$ (with $P_2 \ne A$). The perpendicular from $P_2$ to the line $B_1B_2$ intersects the line $B_1B_2$ at $F$. Prove that if the line $P_1F$ is tangent to $C_1$ then $F$ is the midpoint of the line segment $B_1B_2$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4db59be9fa764d3e910a828ed3296907ca5657.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 1997
[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side.
[b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$.
[b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.)
[b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers.
[b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2015
[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins.
(a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$.
(b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$.
[b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation
$$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012}
= \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$
[b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets.
[b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance.
(a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$.
(b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$.
[b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2006
[b]p1.[/b] In this problem, a half deck of cards consists of $26$ cards, each labeled with an integer from $1$ to $13$. There are two cards labeled $1$, two labeled $2$, two labeled $3$, etc. A certain math class has $13$ students. Each day, the teacher thoroughly shuffles a half deck of cards and deals out two cards to each student. Each student then adds the two numbers on the cards received, and the resulting $13$ sums are multiplied together to form a product $P$. If $P$ is an even number, the class must do math homework that evening. Show that the class always must do math homework.
[b]p2.[/b] Twenty-six people attended a math party: Archimedes, Bernoulli, Cauchy, ..., Yau, and Zeno. During the party, Archimedes shook hands with one person, Bernoulli shook hands with two people, Cauchy shook hands with three people, and similarly up through Yau, who shook hands with $25$ people. How many people did Zeno shake hands with? Justify that your answer is correct and that it is the only correct answer.
[b]p3.[/b] Prove that there are no integers $m, n \ge 1$ such that $$\sqrt{m+\sqrt{m+\sqrt{m+...+\sqrt{m}}}}=n$$ where there are $2006$ square root signs.
[b]p4.[/b] Let $c$ be a circle inscribed in a triangle ABC. Let $\ell$ be the line tangent to $c$ and parallel to $AC$ (with $\ell \ne AC$). Let $P$ and $Q$ be the intersections of $\ell$ with $AB$ and $BC$, respectively. As $ABC$ runs through all triangles of perimeter $1$, what is the longest that the line segment $PQ$ can be? Justify your answer.
[b]p5.[/b] Each positive integer is assigned one of three colors. Show that there exist distinct positive integers $x, y$ such that $x$ and $y$ have the same color and $|x -y|$ is a perfect square.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2011
[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets.
(a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket.
(b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket.
[b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties.
(a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$.
(b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$.
[b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer.
[b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$.
[b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that:
(a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle.
(b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2014
[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same.
(a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square.
(b) Give an example of a multimagic square.
[b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$
Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression.
[b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$
[b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible.
[b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2008
[b]p1.[/b] Show that for every $n \ge 6$, a square in the plane may be divided into $n$ smaller squares, not necessarily all of the same size.
[b]p2.[/b] Let $n$ be the $4018$-digit number $111... 11222...2225$, where there are $2008$ ones and $2009$ twos. Prove that $n$ is a perfect square. (Giving the square root of $n$ is not sufficient. You must also prove that its square is $n$.)
[b]p3.[/b] Let $n$ be a positive integer. A game is played as follows. The game begins with $n$ stones on the table. The two players, denoted Player I and Player II (Player I goes first), alternate in removing from the table a nonzero square number of stones. (For example, if $n = 26$ then in the first turn Player I can remove $1$ or $4$ or $9$ or $16$ or $25$ stones.) The player who takes the last stone wins. Determine if the following sentence is TRUE or FALSE and prove your answer:
There are infinitely many starting values n such that Player II has a winning strategy.
(Saying that Player II has a winning strategy means that no matter how Player I plays, Player II can respond with moves that lead to a win for Player II.)
[b]p4.[/b] Consider a convex quadrilateral $ABCD$. Divide side $AB$ into $8$ equal segments $AP_1$, $P_1P_2$, $...$ , $P_7B$. Divide side $DC$ into $8$ equal segments $DQ_1$, $Q_1Q_2$, $...$ , $Q_7C$. Similarly, divide each of sides $AD$ and $BC$ into $8$ equal segments. Draw lines to form an $8 \times 8$ “checkerboard” as shown in the picture. Color the squares alternately black and white.
(a) Show that each of the $7$ interior lines $P_iQ_i$ is divided into $8$ equal segments.
(b) Show that the total area of the black regions equals the total area of the white regions.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/027f02e26613555181ed93d1085b0e2de43fb6.png[/img]
[b]p5.[/b] Prove that exactly one of the following two statements is true:
A. There is a power of $10$ that has exactly $2008$ digits in base $2$.
B. There is a power of $10$ that has exactly $2008$ digits in base $5$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2016
[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used.
[img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img]
[b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold.
$\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and
$\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$.
[b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin.
[b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer.
[b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties:
$\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and
$\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2021
[b]p1.[/b] The coins in Merryland all have different integer values: there is a single $1$ cent coin, a single $2$ cent coin, etc. What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed $2021$ cents?
[b]p2.[/b] For every positive integer $k$ let $$a_k = \left(\sqrt{\frac{k + 1}{k}}+\frac{\sqrt{k+1}}{k}-\frac{1}{k}-\sqrt{\frac{1}{k}}\right).$$ Evaluate the product $a_4a_5...a_{99}$. Your answer must be as simple as possible.
[b]p3.[/b] Prove that for every positive integer $n$ there is a permutation $a_1, a_2, . . . , a_n$ of $1, 2, . . . , n$ for which $j + a_j$ is a power of $2$ for every $j = 1, 2, . . . , n$.
[b]p4.[/b] Each point of the $3$-dimensional space is colored one of five different colors: blue, green, orange, red, or yellow, and all five colors are used at least once. Show that there exists a plane somewhere in space which contains four points, no two of which have the same color.
[b]p5.[/b] Suppose $a_1 < b_1 < a_2 < b_2 <... < a_n < b_n$ are real numbers. Let $C_n$ be the union of $n$ intervals as below: $$C_n = [a_1, b_1] \cup [a_2, b_2] \cup ... \cup [a_n, b_n].$$
We say $C_n$ is minimal if there is a subset $W$ of real numbers $R$ for which both of the following hold:
(a) Every real number $r$ can be written as $r = c + w$ for some $c$ in $C_n$ and some $w$ in $W$, and
(b) If $D$ is a subset of $C_n$ for which every real number $r$ can be written as $r = d + w$ for some $d$ in $D$ and some $w$ in $W$, then $D = C_n$.
(i) Prove that every interval $C_1 = [a_1, b_1]$ is minimal.
(ii) Prove that for every positive integer $n$, the set $C_n$ is minimal
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Maryland University HSMC part II, 2005
[b]p1.[/b] The three little pigs are learning about fractions. They particularly like the number x = $1/5$, because when they add the denominator to the numerator, add the denominator to the denominator, and form a new fraction, they obtain $6/10$, which equals $3x$ (so each little pig can have his own $x$). The $101$ Dalmatians hear about this and want their own fraction. Your job is to help them.
(a) Find a fraction $y$ such that when the denominator is added to the numerator and also added to the denominator, the result is $101y$.
(b) Prove that the fraction $y$ (put into lowest terms) in part (a) is the only fraction in lowest terms with this property.
[b]p2.[/b] A small kingdom consists of five square miles. The king, who is not very good at math, wants to divide the kingdom among his $9$ sons. He tells each son to mark out a region of $1$ square mile. Prove that there are two sons whose regions overlap by at least $1/9$ square mile.
[b]p3.[/b] Let $\pi (n)$ be the number of primes less than or equal to n. Sometimes $n$ is a multiple of $\pi (n)$. It is known that $\pi (4) = 2$ (because of the two primes $2, 3$) and $\pi (64540) = 6454$. Show that there exists an integer $n$, with $4 < n < 64540$, such that $\pi (n) = n/8$.
[b]p4.[/b] Two circles of radii $R$ and $r$ are externally tangent at a point $A$. Their common external tangent is tangent to the circles at $B$ and $C$. Calculate the lengths of the sides of triangle $ABC$ in terms of $R$ and $r$.
[img]https://cdn.artofproblemsolving.com/attachments/e/a/e5b79cb7c41e712602ec40edc037234468b991.png[/img]
[b]p5.[/b] There are $2005$ people at a meeting. At the end of the meeting, each person who has shaken hands with at most $10$ people is given a red T-shirt with the message “I am unfriendly.” Then each person who has shaken hands only with people who received red T-shirts is given a blue T-shirt with the message “All of my friends are unfriendly.” (Some lucky people might get both red and blue T-shirts, for example, those who shook no one’s hand.) Prove that the number of people who received blue T-shirts is less than or equal to the number of people who received red T-shirts.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].