Let $PT$ and $PB$ be two tangents to a circle, $T$ and $B$ on the circle. $AB$ is the diameter of the circle through $B$ and $TH$ is the perpendicular from $T$ to $AB$, $H$ on $AB$. Prove that $AP$ bisects $TH$.

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.

On the sides $AB$ and $BC$ of triangle $ABC$, there are points $D$ and $E$, respectively, such that \[\angle ADC=\angle BDE\quad\text{and}\quad \angle BCD=\angle AED.\] Prove that $AE=BE$.

If $ \frac {1}{x} \minus{} \frac {1}{y} \equal{} \frac {1}{z}$, then $ z$ equals: $ \textbf{(A)}\ y \minus{} x\qquad \textbf{(B)}\ x \minus{} y\qquad \textbf{(C)}\ \frac {y \minus{} x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y \minus{} x}\qquad \textbf{(E)}\ \frac {xy}{x \minus{} y}$

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.

Given two integers $m\geq 2$ and $n\geq 2$ we consider two types of sequences of length $m\cdot n$ formed exclusively by $0$ and $1$ TYPE 1 sequences are all those that verify the following two conditions: $\bullet$ $a_ka_{k+m} = 0$ for all $k = 1, 2, 3, ...$ $\bullet$ If $a_ka_{k+1} = 1$, then $k$ is a multiple of $m$. TYPE 2 sequences are all those that verify the following two conditions: $\bullet$ $a_ka_{k+n} = 0$ for all $k = 1, 2, 3, ...$ $\bullet$ If $a_ka_{k+1} = 1$, then $k$ is a multiple of $n$. Prove that the number of sequences of type 1 is equal to the number of sequences of type 2.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

Let n be a natural number. Let $A_1, A_2, . . . , A_k$ be distinct $3$-element subsets of $\{1, 2, . . . , n\}$ such that $|A_i \cap A_j | \ne 1$ for all $1 \le i, j \le k$. Determine all $n$ for which there are $n$ such that these subsets exist. [hide=original wording of last sentence]Bestimme alle n, fur die es n solche Teilmengen gibt.[/hide]

Find all positive integers $a$ and $b$ such that \[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \] are both integers.

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: $$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$

[b]p1.[/b] Find all $x$ such that $(\ln (x^4))^2 = (\ln (x))^6$. [b]p2.[/b] On a piece of paper, Alan has written a number $N$ between $0$ and $2010$, inclusive. Yiwen attempts to guess it in the following manner: she can send Alan a positive number $M$, which Alan will attempt to subtract from his own number, which we will call $N$. If $M$ is less than or equal $N$, then he will erase $N$ and replace it with $N -M$. Otherwise, Alan will tell Yiwen that $M > N$. What is the minimum number of attempts that Yiwen must make in order to determine uniquely what number Alan started with? [b]p3.[/b] How many positive integers between $1$ and $50$ have at least $4$ distinct positive integer divisors? (Remember that both $1$ and $n$ are divisors of $n$.) [b]p4.[/b] Let $F_n$ denote the $n^{th}$ Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. Find the last digit of $$\sum^{97!+4}_{i=0}F_i.$$ [b]p5.[/b] Find all prime numbers $p$ such that $2p + 1$ is a perfect cube. [b]p6.[/b] What is the maximum number of knights that can be placed on a $9\times 9$ chessboard such that no two knights attack each other? [b]p7.[/b] $S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers? [b]p8.[/b] In the following infinite array, each row is an arithmetic sequence, and each column is a geometric sequence. Find the sum of the infinite sequence of entries along the main diagonal. [img]https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png[/img] [b]p9.[/b] Let $x > y > 0$ be real numbers. Find the minimum value of $\frac{x}{y} + \frac{4x}{x-y}$ . [b]p10.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$ , for $1 \le i \le 5$ and $1 \le j \le4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p11.[/b] Find the sum of the largest and smallest prime factors of $9^4 + 3^4 + 1$. [b]p12.[/b] A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. Unfortunately, the transmitter makes errors: for each digit in the message, the probability that the transmitter sends the correct digit to the receiver is only $80\%$. (Errors are independent across all digits.) To avoid errors, the receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receiver accepts a message, what is the probability that the message was correct? [b]p13.[/b] Find the integer $N$ such that $$\prod^{8}_{i=0}\sec \left( \frac{\pi}{9}2^i \right)= N.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In quadrilateral $ABCD$, $E$ and $F$ are midpoints of $AB$ and $CD$, and $G$ is the intersection of $AD$ with $BC$. $P$ is a point within the quadrilateral, such that $PA=PB$, $PC=PD$, and $\angle APB+\angle CPD=180^{\circ}$. Prove that $PG$ and $EF$ are parallel.

Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$. Let $F$ be the point of intersection of the lines $BD$ and $CE$. Show that the line $AF$ is perpendicular to the line $BC$.

Let $ABC$ be a triangle, and $\omega_{a}$, $\omega_{b}$, $\omega_{c}$ be circles inside $ABC$, that are tangent (externally) one to each other, such that $\omega_{a}$ is tangent to $AB$ and $AC$, $\omega_{b}$ is tangent to $BA$ and $BC$, and $\omega_{c}$ is tangent to $CA$ and $CB$. Let $D$ be the common point of $\omega_{b}$ and $\omega_{c}$, $E$ the common point of $\omega_{c}$ and $\omega_{a}$, and $F$ the common point of $\omega_{a}$ and $\omega_{b}$. Show that the lines $AD$, $BE$ and $CF$ have a common point.

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

Determine the smallest and the greatest possible values of the expression $$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$. [i]Proposed by Walther Janous, Austria [/i]

In $\triangle ABC$ let point $D$ be the foot of the altitude from $A$ to $BC.$ Suppose that $\angle A = 90^{\circ}, AB - AC = 5,$ and $BD - CD = 7.$ Find the area of $\triangle ABC.$

The exam has $25$ topics, each of which has $8$ questions. On a test, there are $4$ questions of different topics. Is it possible to make $50$ tests so that each question was asked exactly once, and for any two topics there is a test where are questions of both topics?

Given are $m\ge 3$ points in the plane. Prove that you can always choose three of these points $A,B,C$ such that $$\angle ABC \le \frac{180^o}{m}.$$