Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.

Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. What is $m?$ (Recall that two events $A$ and $B$ are independent if $P(A \text{ and }B) = P(A) \cdot P(B).$) $\textbf{(A) } 243 \qquad \textbf{(B) } 245 \qquad \textbf{(C) } 247 \qquad \textbf{(D) } 249\qquad \textbf{(E) } 251$

Let $ABC$ be a triangle with $AB = 3$, $AC = 8$, $BC = 7$ and let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Point $T$ is selected on side $BC$ so that $AT = TC$. The circumcircles of triangles $BAT$, $MAN$ intersect at $D$. Compute $DC$.

Which reaction has the most positive entropy change under standard conditions? $ \textbf{(A)}\hspace{.05in}\ce{H2O}_{(g)}+\ce{CO}_{(g)} \rightarrow \ce{H2}_{(g)}+ \ce{CO2}_{(g)}\qquad$ $\textbf{(B)}\hspace{.05in}\ce{CaCO3}_{(s)} \rightarrow \ce{CaO}_{(s)} + \ce{CO2}_{(g)} \qquad$ $\textbf{(C)}\hspace{.05in}\ce{NH3}_{(g)} \rightarrow \ce{NH3}_{(aq)}\qquad$ $\textbf{(D)}\hspace{.05in}\ce{C8H18}_{(l)} \rightarrow \ce{C8H18}_{(s)}\qquad$

Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)

Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$ [i]Proposed by Alex Schreiber, Germany[/i]

A cyclic quadrilateral $ABCD$ has diameter $AC$ with circumcircle $\omega$. Let $K$ be the foot of the perpendicular from $C$ to $BD$, and the tangent to $\omega$ at $A$ meets $BD$ at $T$. Let the line $AK$ meets $\omega$ at $X$ and choose a point $Y$ on line $AK$ such that $\angle TYA=90^{\circ}$. Prove that $AY=KX$. [i]Proposed by Anzo Teh Zhao Yang[/i]

Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

Fill in the circles to the right with the numbers 1 through 16 so that each number is used once (the number 1 has been filled in already). The number in any non-circular region is equal to the greatest difference between any two numbers in the circles on that region's vertices. You do not need to prove that your configuration is the only one possible; you merely need to find a valid conguration. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [asy] size(190); defaultpen(linewidth(0.8)); int i,j; path p; for(i=0;i<=3;++i){ draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((0,3)--(1,2)^^(0,1)--(2,3)^^(1,0)--(3,2)^^(3,0)--(2,1)); for(i=0;i<=3;++i){ for(j=0;j<=3;++j){ p=circle((i,j),1/4); unfill(p); draw(p); } } label("$1$",(0,3)); label("$7$",(1/3,2+1/3)); label("$8$",(2/3,2+2/3)); label("$2$",(1+1/3,2+2/3)); label("$2$",(1/3,1+2/3)); label("$2$",(2+2/3,1+1/3)); label("$8$",(1+2/3,1/3)); label("$5$",(2+1/3,1/3)); label("$4$",(2+2/3,2/3)); label("$4$",(1/2,1/2)); label("$10$",(3/2,3/2)); label("$11$",(5/2,5/2)); [/asy]

Let $ABC$ be a triangle with $m(\widehat{ABC}) = 90^{\circ}$. The circle with diameter $AB$ intersects the side $[AC]$ at $D$. The tangent to the circle at $D$ meets $BC$ at $E$. If $|EC| =2$, then what is $|AC|^2 - |AE|^2$ ? $\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 12 \qquad\textbf{(E)}\ 10 \qquad\textbf{(E)}\ \text{None}$

Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $$\sum_{i=1}^{2015} f(x_i + x_{i+1}) + f\left( \sum_{i=1}^{2016} x_i \right) \le \sum_{i=1}^{2016} f(2x_i)$$ for all real numbers $x_1, x_2, ... , x_{2016}.$

[b]10.[/b] In an urn there are balls of $N$ different colours, $n$ balls of each colour. Balls are drawn and not replaced until one of the colours turns up twice; denote by $V_{N,n} $ the number of the balls drawn and by $M_{N,n}$ the expectation of the random variable $v_{N,n}$. Find the limit distribution of the random variable $\frac{V_{N,n}}{M_{N,n}}$ if $N \to \infty$ and $n$ is a fixed number. [b](P. 8)[/b]

Let $a,b,c$ be real positive numbers. Prove that the following inequality holds: \[{ \sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2} }\]

Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers

Solve in real numbers the system of equations: \begin{align*} \frac{1}{xy}&=\frac{x}{z}+1 \\ \frac{1}{yz}&=\frac{y}{x}+1 \\ \frac{1}{zx}&=\frac{z}{y}+1 \\ \end{align*}

What is $2021+20+21+2+0+2+1$? [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be an acute triangle with $\angle A < \angle B \le \angle C$, and $O$ its circumcenter. The perpendicular bisector of side $AB$ intersects side $AC$ at $D$. The perpendicular bisector of side $AC$ intersects side $AB$ at $E$. Express the angles of triangle $DEO$ in terms of the angles of triangle $ABC$.

[b]5.[/b] Let $a_0 , a_1 , \dots $ be nonnegative real numbers such that $\sum_{n=0}^{\infty}a_n = \infty$ For arbitrary $ c>0$, let $n_{j}(c)= \min \left \{ k : c.j \leq \sum_{i=0}^{k} a_i \right \}$, $j= 1,2, \dots $ Prove that if $\sum_{i=0}^{\infty}a_i^2 = \infty$, then there exists a $c>0$ for which $\sum_{j=1}^{\infty} a_{n_j (c)} = \infty$ .([b]S.24[/b]) [P. Erdos, I. Joó, L. Székely]

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$

In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$. (Grigory Filippovsky)