Prove that these three statements are equivalent: (a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$. (b) There is no antipodal mapping $f:S^n \to S^{n-1}$. (c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.

Let $n>1$ be an integer. A set $S \subset \{ 0,1,2, \ldots, 4n-1\}$ is called [i]rare[/i] if, for any $k\in\{0,1,\ldots,n-1\}$, the following two conditions take place at the same time (1) the set $S\cap \{4k-2,4k-1,4k, 4k+1, 4k+2 \}$ has at most two elements; (2) the set $S\cap \{4k+1,4k+2,4k+3\}$ has at most one element. Prove that the set $\{0,1,2,\ldots,4n-1\}$ has exactly $8 \cdot 7^{n-1}$ rare subsets.

Alice has $100$ balls and $10$ buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let $b_i$ be the number of balls in the $i$th bucket, for $1\le i \le 10$. Compute the expected value of $\Sigma_{i=1}^{10}b_i^2$

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

Points $A,B,C,D$ are chosen in the plane such that segments $AB,BC,CD,DA$ have lengths $2,7,5,12,$ respectively. Let $m$ be the minimum possible value of the length of segment $AC$ and let $M$ be the maximum possible value of the length of segment $AC.$ What is the ordered pair $(m,M)$?

Each vertex of a regular polygon is colored in one of three colors so that an odd number of vertices are colored in each of the three colors. Prove that the number of isosceles triangles whose vertices are colored in three different colors is odd. [i]From foreign Olympiads[/i]

Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$. [i](3 points)[/i]

p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$ p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$ Find the last seven digits of $A$, in order from millions to units. p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons. p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open? p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.

Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = F C$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$ , and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $Y G$ intersect at $P\ne G$, prove that $GP$ passes through the midpoint of $XY$.

Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$. Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$, such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then: $ \textbf{(A)}\ \text{the triangles are similar in opposite pairs}$ $ \textbf{(B)}\ \text{the triangles are congruent in opposite pairs}$ $ \textbf{(C)}\ \text{the triangles are equal in area in opposite pairs}$ $ \textbf{(D)}\ \text{three similar quadrilaterals are formed}$ $ \textbf{(E)}\ \text{none of the above relations are true}$

Joe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. Then, Joe turns East again and goes to his grandma’s house at $14$ mph. On this last leg, he has to carry flour he bought for her at the store. Her house is $2$ more miles from the store than Joe’s friend’s house is from the store. Joe spends a total of 1 hour on the bike to get to his grandma’s house. If Joe then rides straight home in his grandma’s helicopter at $78$ mph, how many minutes does it take Joe to get home from his grandma’s house

Let $P$ be an arbitrary point of the side line $AB$ of the triangle $ABC$. Mark the perpendicular projection of $P$ on the side lines $AC$ and $BC$ as $A_1$ and $B_1$ respectively. Denote $C_1$ he foot of the alttiude starting from $C$. Prove that the points $A_1$, $B_1$, $C_1$, $C$ and $P$ lie on a circle.

Let $P(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n$ be a polynomial of degree $n$ and $n$ real roots, all of them in the interval $(0,1)$. Prove that for all $k=\overline{1,n}$ the following inequality holds: \[(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.\] [i]Proposed by N. Safaei (Iran)[/i]

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

For any natural $n$, $f(n)$ is the number of labeled digraphs with $n$ vertices such that for any vertex the number if in-edges is equal to the number of out-edges and the total of (in+out) edges is even. Let $g(n)$ be the odd-analogous of $f(n)$. Find $g(n)-f(n)$ with proof . [hide=original formulation] Dado $n$ natural, seja $f(n)$ o número de grafos rotulados direcionados com $n$ vértices de modo que em cada vértice o número de arestas que chegam é igual ao número de arestas que saem e o número de arestas total do grafo é par . Defina $g(n)$ analogamente trocando "par" por "ímpar" na definição acima. Calcule $f(n) - g (n)$. (Observação: Um grafo rotulado direcionado é um par $G = (V, E)$ onde $V = \{1, 2, …, n\}$ e $E$ é um subconjunto de $V^2 -\{(i, i); 0 < i < n + 1\}$).[/hide]

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.

Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome? [i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i]