$A$ is a point lying outside a circle $(O)$. The tangents from $A$ drawn to $(O)$ meet the circle at $B,C.$ Let $P,Q$ be points on the rays $AB, AC$ respectively such that $PQ$ is tangent to $(O).$ The parallel lines drawn through $P,Q$ parallel to $CA, BA,$ respectively meet $BC$ at $E,F,$ respectively. $(a)$ Show that the straight lines $EQ$ always pass through a fixed point $M,$ and $FP$ always pass through a fixed point $N.$ $(b)$ Show that $PM\cdot QN$ is constant.

Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that: \[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\] [i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]

Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$

Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

The first step in finding the product $ (3x \plus{} 2)(x \minus{} 5)$ by use of the distributive property in the form $ a(b \plus{} c) \equal{} ab \plus{} ac$ is: $ \textbf{(A)}\ 3x^2 \minus{} 13x \minus{} 10 \qquad \textbf{(B)}\ 3x(x \minus{} 5) \plus{} 2(x \minus{} 5)\qquad \\\textbf{(C)}\ (3x \plus{} 2)x \plus{} (3x \plus{} 2)( \minus{} 5)\qquad \textbf{(D)}\ 3x^2 \minus{} 17x \minus{} 10\qquad \textbf{(E)}\ 3x^2 \plus{} 2x \minus{} 15x \minus{} 10$

Mr. A owns a home worth $ \$ 10000$. He sells it to Mr. B at a $ 10\%$ profit based on the worth of the house. Mr. B sells the house back to Mr. A at a $ 10\%$ loss. Then: $ \textbf{(A)}\ \text{A comes out even} \qquad\textbf{(B)}\ \text{A makes }\$ 1100\text{ on the deal}\qquad \textbf{(C)}\ \text{A makes }\$ 1000\text{ on the deal}$ $ \textbf{(D)}\ \text{A loses }\$ 900\text{ on the deal} \qquad\textbf{(E)}\ \text{A loses }\$ 1000\text{ on the deal}$

Let be two $ n\times n $ complex matrices $ A,B $ satisfying the equations $ (A+B)^2=A^2+B^2 $ and $ (A+B)^4=A^4+B^4. $ Show that $ (AB)^2=0. $

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?

Prove that for any positive integers $a$, $b$, $c$, at least one of the numbers $a^3b+1$, $b^3c+1$, $c^3a+1$ is not divisible by $a^2+b^2+c^2$.

Let $x,y,z$ be positive real numbers with $x+y+z \ge 3$. Prove that $\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1$ When does equality hold? (Karl Czakler)

Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have: i. $f(f(x,y),z)=f(x,f(y,z))$; ii. $f(x,1)=x, f(1,y)=y$; iii. $f(zx,zy)=z^kf(x,y)$. ($k$ is a positive real number irrelevant to $x,y,z$.)

$n\ge 2$ teams participated in an underwater polo tournament, each two teams played exactly once against each other. A team receives $2, 1, 0$ points for a win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings, the teams were ordered by the total number of points. A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings and ordered them by the total number of points. For which $n$ could the correct order turn out to be the reversed initial order? [i](Proposed by Fedir Yudin)[/i]

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is $\text{(A)}$ an even function, not an odd function. $\text{(B)}$ an odd function, not an even function. $\text{(C)}$ an even function, also an odd function. $\text{(D)}$ neither an even function, nor an odd function.

Let $n$ be a positive integer and $\mathcal{P}_n$ be the set of integer polynomials of the form $a_0+a_1x+\ldots +a_nx^n$ where $|a_i|\le 2$ for $i=0,1,\ldots ,n$. Find, for each positive integer $k$, the number of elements of the set $A_n(k)=\{f(k)|f\in \mathcal{P}_n \}$. [i]Marian Andronache[/i]

Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.

Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that \[ \frac{\log_{10} m}{\log_{10} n}\] is not a rational number.

In $\triangle ABC$, $\angle B=90^\circ$, segment $AB>BC$. Now we have a $\triangle A_iBC(i=1,2,...,n)$ which is similiar to $\triangle ABC$ (the vertexs of them might not correspond). Find the maximum value of $n+2018$.

Prove that for any four real numbers $a$, $b$, $c$, $d$, the inequality \[ \left(a-b\right)\left(b-c\right)\left(c-d\right)\left(d-a\right)+\left(a-c\right)^2\left(b-d\right)^2\geq 0 \] holds. [hide="comment"]This is inequality (350) in: Mihai Onucu Drimbe, [i]Inegalitati, idei si metode[/i], Zalau: Gil, 2003. Posted here only for the sake of completeness; in fact, it is more or less the same as http://www.mathlinks.ro/Forum/viewtopic.php?t=3152 .[/hide] Darij

Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have $$2f(ab)=(b+1)f(a)+(a+1)f(b).$$ Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that $$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$ where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime. [i]Authored by Nikola Velov[/i]

Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \] a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

In trapezoid $ ABCD$ we have $ \overline{AB}$ parallel to $ \overline{DC}$, $ E$ as the midpoint of $ \overline{BC}$, and $ F$ as the midpoint of $ \overline{DA}$. The area of $ ABEF$ is twice the area of $ FECD$. What is $ AB/DC$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.