$a,b$ are two different positive real numbers, then which one is the largest? $$A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.$$ $\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}$Not sure.

Consider the polynomial $p(x) = x^4 + ax^3 + bx^2 + cx + d$. It is given that $p(x)$ has its only root at $x = r$ i.e $p(r) = 0$. $\textbf{(a)}$ Show that if $a, b, c, d$ are rational then $r$ is rational. $\textbf{(b)}$ Show that if $a, b, c, d$ are integers then $r$ is an integer. [hide=Hint](Hint: Consider the roots of $p'(x)$ )[/hide]

Let $\triangle ABC$ be an acute triangle, and let $D,E,F$ respectively be three points on sides $BC,CA,AB$ such that $AEDF$ is a cyclic quadrilateral. Let $O_B$ and $O_C$ be the circumcenters of $\triangle BDF$ and $\triangle CDE$, respectively. Finally, let $D'$ be a point on segment $BC$ such that $BD'=CD$. Prove that $\triangle BD'O_B$ and $\triangle CD'O_C$ have the same surface.

Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$

1) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy? 2) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b+2$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?

Let $\mathcal F$ be the set of all continuous functions $f:[0,1]\to\mathbb R$ with the property $$\left|\int^x_0\frac{f(t)}{\sqrt{x-t}}\text dt\right|\le1\enspace\text{for all }x\in(0,1].$$Compute $\sup_{f\in\mathcal F}\left|\int^1_0f(x)\text dx\right|$.

Given a regular nonagon $A_1A_2A_3A_4A_5A_6A_7A_8A_9$ with side length $1$. Diagonals $A_3A_7$ and $A_4A_8$ intersect at point $P$. Find the length of segment $P A_1$.

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

Let $a,b,c$ be three integers for which the sum \[ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\] is integer. Prove that each of the three numbers \[ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}\] is integer. (Proposed by Gerhard J. Woeginger)

Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.

Compute $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{8 \cdot 4^n - 6 \cdot 2^n +1}$$

Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? $\textbf{(A)}\ 1:24\qquad \textbf{(B)}\ 1:28\qquad \textbf{(C)}\ 1:36\qquad \textbf{(D)}\ 1:40\qquad \textbf{(E)}\ 1:48$

You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ [i]good[/i] if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane. [i]Proposed by Anton Trygub[/i]

Square $ABCD$ has side length $4$. Points $P$ and $Q$ are located on sides $BC$ and $CD$, respectively, such that $BP = DQ = 1$. Let $AQ$ intersect $DP$ at point $X$. Compute the area of triangle $P QX$.

Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$ [i](A. Golovanov )[/i]

Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$

$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}$$ Prove that $h$ is differentiable and find $h'$. \\ \\ $\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $. \\ \\ [i](Sergiu Novac)[/i]

For $x$ real, the inequality $1\le |x-2|\le 7$ is equivalent to $\textbf{(A) }x\le 1\text{ or }x\ge 3\qquad\textbf{(B) }1\le x\le 3\qquad\textbf{(C) }-5\le x\le 9\qquad$ $\textbf{(D) }-5\le x\le 1\text{ or }3\le x\le 9\qquad \textbf{(E) }-6\le x\le 1\text{ or }3\le x\le 10$

Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas. What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?

Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select? $\textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{5}{36} \qquad\textbf{(C)}\ \frac{14}{45} \qquad\textbf{(D)}\ \frac{25}{63} \qquad\textbf{(E)}\ \frac{1}{2}$

Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

Let $\mathcal P$ be a finite set of squares on an infinite chessboard. Kelvin the Frog notes that $\mathcal P$ may be tiled with only $1 \times 2$ dominoes, while Alex the Kat notes that $\mathcal P$ may be tiled with only $2 \times 1$ dominoes. The dominoes cannot be rotated in each tiling. Prove that the area of $\mathcal P$ is a multiple of 4.

Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.

Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$. [i]2016 CCA Math Bonanza Individual #13[/i]

Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$ b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$