Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$. Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.

Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: [list=disc] [*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests. [/list] Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.

Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$

Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.

A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be [i]happy[/i] if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

Prove that: [b](a)[/b] $ 5<\sqrt {5}\plus{}\sqrt [3]{5}\plus{}\sqrt [4]{5}$ [b](b)[/b] $ 8>\sqrt {8}\plus{}\sqrt [3]{8}\plus{}\sqrt [4]{8}$ [b](c)[/b] $ n>\sqrt {n}\plus{}\sqrt [3]{n}\plus{}\sqrt [4]{n}$ for all integers $ n\geq 9 .$ [b][Weightage 16/100][/b]

Let $X_1$, $Z_2$, $Y_1$, $X_2$, $Z_1$, $Y_2$ be six points lying on the periphery of a circle (in this order). Let the chords $Y_1Y_2$ and $Z_1Z_2$ meet at a point $A$; let the chords $Z_1Z_2$ and $X_1X_2$ meet at a point $B$; let the chords $X_1X_2$ and $Y_1Y_2$ meet at a point $C$. Prove that $\left( BX_2-CX_1\right) \cdot BC+\left( CY_2-AY_1\right) \cdot CA+\left( AZ_2-BZ_1\right) \cdot AB=0$. [i]Comment on the source.[/i] The problem is inspired by Stergiu's proof in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=326112#p326112]http://www.mathlinks.ro/Forum/viewtopic.php?t=50262 post #5[/url]. Darij

Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points. Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong). (a) Does $\sum_{i=1}^{p-1}\frac{1}{i}\equiv 0\pmod{p^2}$ for all odd prime numbers $p$? (Note that $\frac{1}{i}$ denotes the number such that $i\cdot\frac{1}{i}\equiv 1\pmod{p^2}$) (b) Do there exist $2017$ positive perfect cubes that sum to a perfect cube? (c) Does there exist a right triangle with rational side lengths and area $5$? (d) A [i]magic square[/i] is a $3\times 3$ grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all [color = red]different[/color] prime numbers? (e) Is $\prod_{p} \frac{p^2+1}{p^2-1} = \frac{2^2+1}{2^2-1}\cdot\frac{3^2+1}{3^2-1}\cdot\frac{5^2+1}{5^2-1}\cdot\frac{7^2+1}{7^2-1}\cdot\dots$ a rational number? (f) Do there exist infinite number of pairs of [i]distinct[/i] integers $(a,b)$ such that $a$ and $b$ have the same set of prime divisors, and $a+1$ and $b+1$ have the same set of prime divisors? [color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color] [color = red]A clarification was issued for problem 36(d) during the test. I have included it above.[/color]

Of the following statements, the only one that is incorrect is: $ \textbf{(A)}$ An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity. $ \textbf{(B)}$ The arithmetic mean of two unequal positive quantities is greater than their geometric mean. $ \textbf{(C)}$ If the sum of two positive quantities is given, ther product is largest when they are equal. $ \textbf{(D)}$ If $ a$ and $ b$ are positive and unequal, $ \frac {1}{2}(a^2 \plus{} b^2)$ is greater than $ [\frac {1}{2}(a \plus{} b)]^2$. $ \textbf{(E)}$ If the product of two positive quantities is given, their sum is greatest when they are equal.

Suppose $2$ cars are going into a turn the shape of a half-circle. Car $ 1$ is traveling at $50$ meters per second and is hugging the inside of the turn, which has radius $200$ meters. Car $2$ is trying to pass Car $ 1$ going along the turn, but in order to do this, he has to move to the outside of the turn, which has radius $210$. Suppose that both cars come into the turn side by side, and that they also end the turn being side by side. What was the average speed of Car $2$, in meters per second, throughout the turn?

Let $m, n$ be positive integers and $a_1,\dots , a_m, b_1, \dots , b_n$ positive real numbers such that for every positive integer $k$ we have that $$(a_1^k + \cdots + a^k_m) - (b^k_1 + \cdots + b^k_n) \leq CkN, $$ for some fix $C$ and $N$. Show that there exists $l \leq m, n$ and permutations $\sigma$ of $\{1, \dots , m\}$ and $\tau$ of $\{1,\dots , n\}$, such that 1. $a\sigma(i) = b\tau(i)$ for $1 \leq i \leq l,$ 2. $a\sigma(i) , b\tau(i) \leq 1$ for $i > l.$

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: [center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center] [center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center] Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy. [i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$. [i]Proposed by M. Panov[/i]

For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$[i]th finite difference[/i], denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by \begin{align*} \Delta^0 f(n)&=f(n)\\ \Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots. \end{align*} Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$. [i]Holden Mui[/i]

What is the minimum number of cells that can be colored black in white square $ 300 \times 300 $ so that no three black cells formed a corner, and after painting any white cell this condition violated?

Two rays start from a common point and have an angle of $60$ degrees. Circle $C$ is drawn with radius $42$ such that it is tangent to the two rays. Find the radius of the circle that has radius smaller than circle $C$ and is also tangent to $C$ and the two rays.

Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative'' sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$. [i] Proposed by Ashwin Sah [/i]

Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.

Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.