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)

A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point.

[b]9.[/b] Spin a regular coin repeatedly until the heads and tails turned up both reach the number $k$ ($k= 1, 2, \dots $); denote by $v_k$ the number of the necessary throws. Find the distribution of the random variable $v_k$ and the limit-distribution of the random variable $\frac {v_k -2k}{\sqrt {2k}}$ as $k \to \infty$. [b](P. 10)[/b]

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.

Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number \[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\] is a multiple of $19$.

Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively Find the angles of the triangle $ABC$