Find all sets $ A$ of nonnegative integers with the property: if for the nonnegative intergers $m$ and $ n $ we have $m+n\in A$ then $m\cdot n\in A.$

$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value, a) Prove that $ AB \geq 2BC$, b) Find the value of $ AQ \cdot BQ$.

$f$ be a twice differentiable real valued function satisfying \[ f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) \] where $g(x)\ge 0$ for all real $x$. Show that $|f(x)|$ is bounded.

Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices. $\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$

Five points $A, B, C, D$, and $E$ in three-dimensional Euclidean space have the property that $AB = BC = CD = DE = EA = 1$ and $\angle ABC = \angle BCD =\angle CDE = \angle DEA = 90^o$ . Find all possible $\cos(\angle EAB)$.

Consider the sequence $\{a_n\}$ defined so that $a_n$ is the leftmost digit of $2^n$. The first few terms of this sequence are $1,2,4,8,1,3,6,\dots$. For how many $0\le n\le100000$ is $a_n=1$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)$.

We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?

In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's?

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. Find the ratio $L/d$. $ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

The geometric mean of a set of $m$ non-negative numbers is the $m$-th root of the product of these numbers. For which positive values of ​​$n$, is there a finite set $S_n$ of $n$ positive integers different such that the geometric mean of any subset of $S_n$ is an integer?

Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test? $\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$

The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: $\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad \textbf{(B)}\ f(n)\qquad \textbf{(C)}\ 2f(n)+1 \qquad \textbf{(D)}\ f^2(n) \qquad \textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

Let $a\in (0,1)$, $f(z)=z^2-z+a, z\in \mathbb{C}$. Prove the following statement holds: For any complex number z with $|z| \geq 1$, there exists a complex number $z_0$ with $|z_0|=1$, such that $|f(z_0)| \leq |f(z)|$.

Let \( n \geq 2 \) be an integer. Two players, Alice and Bob, play the following game on the complete graph \( K_n \): They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored. Prove that there exists a positive number \(\varepsilon\), Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding \[ \left( \frac{1}{4} - \varepsilon \right) n^2 + n. \]

Quadrilateral $ABCD$ has integer side lengths, and angles $ABC, ACD$, and $BAD$ are right angles. Compute the smallest possible value of $AD$.

The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles? [asy] unitsize(45); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0)); dot((0,0),ds); label("$A$", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label("$B$", (1.97,-0.31),NE*lsf); dot((2,1),ds); label("$C$", (1.96,1.09),NE*lsf); dot((4,0),ds); label("$D$", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy] $ \textbf{(A)}\ 36 \pi \qquad\textbf{(B)}\ 49 \pi\qquad\textbf{(C)}\ 64 \pi\qquad\textbf{(D)}\ 81 \pi\qquad\textbf{(E)}\ 100 \pi $

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $1$. Let the unit circles centered at $A$, $B$, and $C$ be $\Omega_A$, $\Omega_B$, and $\Omega_C$, respectively. Then, let $\Omega_A$ and $\Omega_C$ intersect again at point $D$, and $\Omega_B$ and $\Omega_C$ intersect again at point $E$. Line $BD$ intersects $\Omega_B$ at point $F$ where $F$ lies between $B$ and $D$, and line $AE$ intersects $\Omega_A$ at $G$ where $G$ lies between $A$ and $E$. $BD$ and $AE$ intersect at $H$. Finally, let $CH$ and $FG$ intersect at $I$. Compute $IH$. [b]p14.[/b] Suppose Bob randomly fills in a $45 \times 45$ grid with the numbers from $1$ to $2025$, using each number exactly once. For each of the $45$ rows, he writes down the largest number in the row. Of these $45$ numbers, he writes down the second largest number. The probability that this final number is equal to $2023$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Compute the value of $p$. [b]p15.[/b] $f$ is a bijective function from the set $\{0, 1, 2, ..., 11\}$ to $\{0, 1, 2, ... , 11\}$, with the property that whenever $a$ divides $b$, $f(a)$ divides $f(b)$. How many such $f$ are there? [i]A bijective function maps each element in its domain to a distinct element in its range. [/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has $AB=13$, $BC=14$, and $CA=15$. Let $\omega_A$, $\omega_B$ and $\omega_C$ be circles such that $\omega_B$ and $\omega_C$ are tangent at $A$, $\omega_C$ and $\omega_A$ are tangent at $B$, and $\omega_A$ and $\omega_B$ are tangent at $C$. Suppose that line $AB$ intersects $\omega_B$ at a point $X \neq A$ and line $AC$ intersects $\omega_C$ at a point $Y \neq A$. If lines $XY$ and $BC$ intersect at $P$, then $\tfrac{BC}{BP} = \tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Ren[/i]

Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.