Let $n{}$ be a positive integer, and let $\mathcal{C}$ be a collection of subsets of $\{1,2,\ldots,2^n\}$ satisfying both of the following conditions:[list=1] [*]Every $(2^n-1)$-element subset of $\{1,2,\ldots,2^n\}$ is a member of $\mathcal{C}$, and [*]Every non-empty member $C$ of $\mathcal{C}$ contains an element $c$ such that $C\setminus\{c\}$ is again a member of $\mathcal{C}$. [/list]Determine the smallest size $\mathcal{C}$ may have. [i]Serbia, Pavle Martinovic ́[/i]

Solve $|x-1|-2|x+5|>3+x$.

There are $mn$ line segments in a plane that connect $n$ given points. Prove that a sequence $V_0$, $V_1$, $...$, $V_m$ of different points can be selected from them such that $V_{i-1}$ and $V_i$ are connected by a line ($1 \le i \le m$).

For any angle α with $0 < \alpha < 180^{\circ}$, we call a closed convex planar set an $\alpha$-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$. Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$-set $S \subset T .$

Fill each unshaded cell of the grid with a number that is either $1$, $3$, or $5$. For each cell, exactly one of the touching cells must contain the same number. Here touching includes cells that only share a point, i.e. touch diagonally. [asy] unitsize(1.2cm); defaultpen(fontsize(30pt)); for(int i=0; i<8; ++i){ draw((i,0)--(i,7)); draw((0,i)--(7,i)); } filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray); filldraw((1,1)--(1,2)--(2,2)--(2,1)--cycle, gray); filldraw((4,2)--(4,3)--(5,3)--(5,2)--cycle, gray); filldraw((3,4)--(4,4)--(4,5)--(3,5)--cycle, gray); filldraw((6,2)--(6,3)--(7,3)--(7,2)--cycle, gray); int[][] numbers = { {0,0,0,0,0,0,0}, {1,0,1,0,3,0,5}, {0,0,0,0,0,0,0}, {0,0,1,1,3,5,0}, {0,0,0,0,0,0,0}, {0,0,0,0,0,0,0}, {5,0,0,5,0,0,0}}; for(int i=0; i<7; ++i){ for(int j=0; j<7; ++j){ if(numbers[i][j]>0){label(string(numbers[i][j]),(j+0.5,6.5-i));} } } [/asy] There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the conditions of the problem. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

Consider an arithmetic progression made up of $100$ terms. If the sum of all the terms of the progression is $150$ and the sum of the even terms is $50$, find the sum of the squares of the $100$ terms of the progression.

Denote by $S(x)$ the sum of digits of positive integer $x$ written in decimal notation. For $k$ a fixed positive integer, define a sequence $(x_n)_{n \geq 1}$ by $x_1=1$ and $x_{n+1}$ $=$ $S(kx_n)$ for all positive integers $n$. Prove that $x_n$ $<$ $27 \sqrt{k}$ for all positive integer $n$.

Given that $a>2b$ and $b>2c$ and $a$, $b$, and $c$ are nonzero, which of the following statements must be true? $\textbf{(A) } a+b>c\qquad\textbf{(B) } a-c>0\qquad\textbf{(C) } abc>0\qquad\textbf{(D) } \frac{a}{b}>2\qquad\textbf{(E) } \text{none of these}$

Let $a,b,c\in Z$ and $a$ be the even number and $b$ be the odd number. Show that for every integer $n$ there exist one positive integer $x$ such that $2^n\mid ax^2+bx+c$

Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.

A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?

Points $A, B$ are given. Find the locus of points $C$ such that $C$, the midpoints of $AC, BC$ and the centroid of triangle $ABC$ are concyclic.

Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$ for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$. [i]Proposed by Wong Jer Ren[/i]

What is the units digit of $19^{19} + 99^{99}$? $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Given positive integers $d \ge 3$, $r>2$ and $l$, with $2d \le l <rd$. Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$, and no more than $l-d$. A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$, such that for $G$'s any proper coloring with $r-1$ colors, and for an arbitrary color $C$, either all numbers in color $C$ appear in $A$, or none of the numbers in color $C$ appear in $A$. Show that $G$ has a proper coloring within $r-1$ colors.

Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.

$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them.

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$.

For a triangle $ ABC $ which $ \angle B \ne 90^{\circ} $ and $ AB \ne AC $, define $ P_{ABC} $ as follows ; Let $ I $ be the incenter of triangle $ABC$, and let $ D, E, F $ be the intersection points with the incircle and segments $ BC, CA, AB $. Two lines $ AB $ and $ DI $ meet at $ S $ and let $ T $ be the intersection point of line $ DE $ and the line which is perpendicular with $ DF $ at $ F $. The line $ ST $ intersects line $ EF $ at $ R$. Now define $ P_{ABC} $ be one of the intersection points of the incircle and the circle with diameter $ IR $, which is located in other side with $ A $ about $ IR $. Now think of an isosceles triangle $ XYZ $ such that $ XZ = YZ > XY $. Let $ W $ be the point on the side $ YZ $ such that $ WY < XY $ and Let $ K = P_{YXW} $ and $ L = P_{ZXW} $. Prove that $ 2 KL \le XY $.

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

Find all rectangles that can be cut into $13$ equal squares.

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

Are there exist some positive integers $n$ and $k$, such that the first decimals of $2^n$ (from left to the right) represent the number $5^k$ while the first decimals of $5^n$ represent the number $2^k$ ? (5)