Let $E$ and $F$ be points on the side $AB$ of a rectangle $ABCD$ such that $AE = EF$. The line through $E$ perpendicular to $AB$ intersects the diagonal $AC$ at $G$, and the segments $FD$ and $BG$ intersect at $H$. Prove that the areas of the triangles $FBH$ and $GHD$ are equal.

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

Suppose that $a$ and $b$ are two distinct positive real numbers such that $\lfloor na\rfloor$ divides $\lfloor nb\rfloor$ for any positive integer $n$. Prove that $a$ and $b$ are positive integers.

Three given circles have the same radius and pass through a common point $ P$. Their other points of pairwise intersections are $ A$, $ B$, $ C$. We define triangle $ A'B'C'$, each of whose sides is tangent to two of the three circles. The three circles are contained in $ \triangle A'B'C'$. Prove that the area of $ \triangle A'B'C'$ is at least nine times the area of $ \triangle ABC$

What is the maximum number of triangles with vertices on the points of the fork/graph which is possible to construct?

Let $R $ be the circumradius of a circumscribed quadrilateral $ABCD $. Let $h_1$ and $h_2$ be the altitudes from $A $ to $BC $ and $CD $ respectively. Similarly $h_3$ and $h_4$ are the altitudes from $C $ to $AB $ and $AD$. Prove that $$\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4} $$

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$. [b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer. [b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy. [b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers. [b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$. [b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$ where $m|496$ means $496$ is divisible by $m$. [b]p7.[/b] What is the value of $${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$ [b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function $$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$ Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

The sum of eleven consecutive integers is $2002$. What is the smallest of these integers? $\textbf{(A) }175\qquad\textbf{(B) }177\qquad\textbf{(C) }179\qquad\textbf{(D) }180\qquad\textbf{(E) }181$

For a positive integer $n \ge 5$, let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the following two conditions: [list] (a) The pairs $(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$; (b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$. [/list] Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$.

In triangle $ABC$ such that $\angle ACB=90^{\circ}$, let point $H$ be foot of perpendicular from point $C$ to side $AB$. Show that sum of radiuses of incircles of $ABC$, $BCH$ and $ACH$ is $CH$

Let \( M \) be the midpoint of side \( BC \) of a scalene triangle \( ABC \). The circle passing through \( A \), \( B \) and \( M \) intersects side \( AC \) again at \( D \). The circle passing through \( A \), \( C \) and \( M \) cuts side \( AB \) again at \( E \). Let \( O \) be the circumcenter of triangle \( ADE \). Prove that \( OB=OC \).

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$?

Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.

There are 7 lines in the plane. A point is called a [i]good[/i] point if it is contained on at least three of these seven lines. What is the maximum number of [i]good[/i] points?

A point $ P$ in the interior of triangle $ ABC$ satisfies \[ \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB.\] Prove that \[ \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.\]

For positive integers $m$ and $n$, let $r(m, n)$ be the remainder when $m$ is divided by $n$. Find the smallest positive integer $m$ such that \[r(m, 1) + r(m, 2) + r(m, 3) +\cdots+ r(m, 10) = 4.\]

In convex quadrilateral $ABCD$ with $AB = 11$ and $CD = 13,$ there is a point $P$ for which $\triangle{ADP}$ and $\triangle{BCP}$ are congruent equilateral triangles. Compute the side length of these triangles.

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]