In the adjoining figure $ TP$ and $ T'Q$ are parallel tangents to a circle of radius $ r$, with $ T$ and $ T'$ the points of tangency. $ PT''Q$ is a third tangent with $ T''$ as point of tangency. If $ TP\equal{}4$ and $ T'Q\equal{}9$ then $ r$ is [asy]unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label("O",O,W); label("T",T,N); label("T'",T1,S); label("T''",T2,NE); label("P",P,NE); label("Q",Q,S); draw(O--T2); label("$r$",midpoint(O--T2),NW); draw(T--P); label("4",midpoint(T--P),N); draw(T1--Q); label("9",midpoint(T1--Q),S); draw(P--Q);[/asy] $ \textbf{(A)}\ 25/6 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 25/4 \\ \qquad \textbf{(D)}\ \text{a number other than }25/6, 6, 25/4 \\ \qquad \textbf{(E)}\ \text{not determinable from the given information}$

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? $ \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 $

The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, then $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and $ x \oplus y$ denotes the concatenation of x and y. The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ is defined as above and $ x \oplus y$ is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x. Prove that the two languages consist of the same words.

Points $A,B,C$, and $M$ are given in the plane. (a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$. (b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

The top of one tree is $16$ feet higher than the top of another tree. The height of the $2$ trees are at a ratio of $3:4$. In feet, how tall is the taller tree? $ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112 $

In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 $

Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$? $ \textbf{(A)}\ \frac78 \qquad\textbf{(B)}\ \frac{\sqrt 3}2 \qquad\textbf{(C)}\ \frac 87 \qquad\textbf{(D)}\ \frac 38 \qquad\textbf{(E)}\ \frac{\sqrt 2}2 $

How many three-digit numbers have at least one $2$ and at least one $3$? $\textbf{(A) }52\qquad\textbf{(B) }54\qquad\textbf{(C) }56\qquad\textbf{(D) }58\qquad\textbf{(E) }60$

Let the quadratic $P(x)=x^2+5x+1$. Two distinct real numbers $a,b$ satisfy \[P(a+b)=ab\] \[P(ab)=a+b\] Find the sum of all possible values of $a^2$. [i]Proposed by Harry Kim[/i]

Evaluate the expression \[ \left| \prod_{k=0}^{15} \left( 1+e^{2\pi i k^2/{31}} \right) \right| \, . \]

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Find the value of the sum \[ \left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor+\left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor+\cdots+\left\lfloor 2+\frac{1}{2^1} \right\rfloor+\left\lfloor 2+\frac{1}{2^0} \right\rfloor. \]

On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.

Suppose $\omega$ is a circle centered at $O$ with radius $8$. Let $AC$ and $BD$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $ABCD$ such that the circumcircles of triangles $ABP$ and $CDP$ are tangent, and the circumcircles of triangles $ADP$ and $BCP$ are tangent. If $AC = 2\sqrt{61}$ and $BD = 6\sqrt7$,then $OP$ can be expressed as $\sqrt{a}-\sqrt{b}$ for positive integers $a$ and $b$. Compute $100a + b$.

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations.

There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$ \[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

$ ABCD$ is a square. Let $ E$ be a point on the segment $ BC$ and $ F$ be a point on the segment $ ED$. If $ DF \equal{} BF$ and $ EF \equal{} BE$, then $ \angle DFA$ is $\textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 60^\circ \qquad\textbf{(C)}\ 75^\circ \qquad\textbf{(D)}\ 80^\circ \qquad\textbf{(E)}\ 85^\circ$

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

A poll shows that $ 70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? $ \textbf{(A)}\ 0.063 \qquad \textbf{(B)}\ 0.189 \qquad \textbf{(C)}\ 0.233 \qquad \textbf{(D)}\ 0.333 \qquad \textbf{(E)}\ 0.441$

In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores: (a) Michael K had an average test score of $90$, Michael M had an average test score of $91$, and Michael R had an average test score of $92$. (b) Michael K took more tests than Michael M, who in turn took more tests than Michael R. (c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests) What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined? [i]Proposed by James Lin[/i]