A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis. Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.

Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.

In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.

Let there be $(a_n)_{n\ge1},(b_n)_{n\ge1},a_n,b_n\in\mathbb R^*_+=(0,\infty)$ such that $\lim_{n\to\infty}a_n=a\in\mathbb R^*_+$ and $(b_n)_{n\ge1}$ is a bounded sequence. If $(x_n)_{n\ge1}$, $x_n=\prod_{k=1}^n(ka_h+b_h)$ find: $$\lim_{n\to\infty}\left(\sqrt[n+1]{x_{n+1}}-\sqrt[n]{x_n}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.

The minimum value of $ x(x \plus{} 4)(x \plus{} 8)(x \plus{} 12)$ in real numbers is ? $\textbf{(A)}\ \minus{} 240 \qquad\textbf{(B)}\ \minus{} 252 \qquad\textbf{(C)}\ \minus{} 256 \qquad\textbf{(D)}\ \minus{} 260 \qquad\textbf{(E)}\ \minus{} 280$

Let $n$ be a positive integer. Prove that \[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]

An illusionist and his assistant are about to perform the following magic trick. Let $k$ be a positive integer. A spectator is given $n=k!+k-1$ balls numbered $1,2,…,n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as he sees fit. The assistant studies the sequence, chooses some block of $k$ consecutive balls, and covers them under her scarf. Then the illusionist looks at the newly obscured sequence and guesses the precise order of the $k$ balls he does not see. Devise a strategy for the illusionist and the assistant to follow so that the trick always works. (The strategy needs to be constructed explicitly. For instance, it should be possible to implement the strategy, as described by the solver, in the form of a computer program that takes $k$ and the obscured sequence as input and then runs in time polynomial in $n$. A mere proof that an appropriate strategy exists does not qualify as a complete solution.)

$P(x)$ is polynomial with degree $n>5$ and integer coefficients have $n$ different integer roots. Prove that $P(x)+3$ have $n$ different real roots.

The value of $x$ in the interval $[0, 2\pi]$ that minimizes the value of $x + 2\cos x$ can be written in the form $a\pi/b,$ where $a$ and $b$ are relatively prime positive integers. Compute $a + b.$

Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?

Prove that the equation \[\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] has infinitly many natural solutions

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions: i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$ ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$ Determine the minimum value of $m$.

There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.

Fill in each cell of the grid with a positive digit so that the following conditions hold: 1. each row and column contains ve distinct digits; 2. for any cage containing multiple cells of a row, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole row, and 3. for any cage containing multiple cells of a column, the label on the cage is the GCD of the sum of the digits in the cage and the sum of the digits in the whole column. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (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] unitsize(48); int[][] a = { {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}; for (int i = 0; i < 5; ++i) { for (int j = 0; j < 5; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0 && a[j][i] < 999) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(24pt)); } } real ep=0.1; real s=3; pen lw=linewidth(.12mm); real x=0.9; real y=1.2; draw((0+s*ep,0-ep)--(2-ep,0-ep)--(2-ep,-1+ep)--(0+ep,-1+ep)--(0+ep,0-s*ep),dashed+lw); draw((2+s*ep,0-ep)--(4-ep,0-ep)--(4-ep,-1+ep)--(2+ep,-1+ep)--(2+ep,0-s*ep),dashed+lw); draw((1+s*ep,-1-ep)--(3-ep,-1-ep)--(3-ep,-2+ep)--(1+ep,-2+ep)--(1+ep,-1-s*ep),dashed+lw); draw((2+s*ep,-3-ep)--(4-ep,-3-ep)--(4-ep,-4+ep)--(2+ep,-4+ep)--(2+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-4-ep)--(3-ep,-4-ep)--(3-ep,-5+ep)--(1+ep,-5+ep)--(1+ep,-4-s*ep),dashed+lw); draw((3+s*ep,-4-ep)--(5-ep,-4-ep)--(5-ep,-5+ep)--(3+ep,-5+ep)--(3+ep,-4-s*ep),dashed+lw); label(scale(x)*"5", (0+ep,0-y*ep)); label(scale(x)*"7", (2+ep,0-y*ep)); label(scale(x)*"10", (1+ep,-1-y*ep)); label(scale(x)*"5", (2+ep,-3-y*ep)); label(scale(x)*"2", (1+ep, -4-y*ep)); label(scale(x)*"13", (3+ep, -4-y*ep)); draw((4+s*ep,0-ep)--(5-ep,0-ep)--(5-ep,-2+ep)--(4+ep,-2+ep)--(4+ep,0-s*ep),dashed+lw); draw((0+s*ep,-1-ep)--(1-ep,-1-ep)--(1-ep,-3+ep)--(0+ep,-3+ep)--(0+ep,-1-s*ep),dashed+lw); draw((3+s*ep,-1-ep)--(4-ep,-1-ep)--(4-ep,-3+ep)--(3+ep,-3+ep)--(3+ep,-1-s*ep),dashed+lw); draw((0+s*ep,-3-ep)--(1-ep,-3-ep)--(1-ep,-5+ep)--(0+ep,-5+ep)--(0+ep,-3-s*ep),dashed+lw); draw((1+s*ep,-2-ep)--(2-ep,-2-ep)--(2-ep,-4+ep)--(1+ep,-4+ep)--(1+ep,-2-s*ep),dashed+lw); draw((4+s*ep,-2-ep)--(5-ep,-2-ep)--(5-ep,-4+ep)--(4+ep,-4+ep)--(4+ep,-2-s*ep),dashed+lw); label(scale(x)*"10", (4+ep,0-y*ep)); label(scale(x)*"3", (0+ep,-1-y*ep)); label(scale(x)*"8", (3+ep,-1-y*ep)); label(scale(x)*"16", (1+ep,-2-y*ep)); label(scale(x)*"6", (4+ep,-2-y*ep)); label(scale(x)*"11", (0+ep,-3-y*ep)); [/asy]

Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$

ff nonnegative real numbers$ x_1,x_2,...,x_n$ satisfy $x_1 +...+x_n\le \frac12$, prove that $$(1-x_1)(1-x_2)...(1-x_n) \ge \frac12$$

The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that $AB$ goes to $DA$ $DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$ then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?

A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule \[a_{n+1}=a_n+b_n\ (n=1,2,\ldots)\] where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.

Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$. [i]V. Kamianetski[/i]

A Tetris Figure is every figure in the plane which consists of $4$ unit squares connected by their sides (and don’t overlap). Two Tetris Figures are the same if one can be rotated in the plane to become the other. a) Prove that there exist exactly $7$ different Tetris Figures. b) Is it possible to fill a $4 \times 7$ rectangle by using once each of the $7$ different Tetris Figures?

For $p$, $q$, $l$ strictly positive real numbers, consider the following problem: for $y \geq 0$ fixed, determine the values $x \geq 0$ such that $x^p - lx^q \leq y$. Denote by $S(y)$ the set of solutions of this problem. Prove that if one has $p < q$, $\epsilon \in (0, l^\frac{1}{p-q})$, $0 \leq x \leq \epsilon$ and $x \in S(y)$, then $$x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}$$

Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]