Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]

Let $P$ and $Q$ be non-constant integer-coefficient monic polynomials, and let $a$ and $b$ be integers satisfying $| a | \geq 3$ and $ | b | \geq 3$. These satisfy the following conditions for all positive integers $n$: $$ P(n) \mid Q(n)^2 + aQ(n) + 1, \quad Q(n) \mid P(n)^2 + bP(n) + 1. $$ Determine all possible ordered pairs $(a+b, \deg P)$. [hide=Original wording] 상수다항식이 아닌 최고차항의 계수가 1인 정수계수다항식 $P$, $Q$와 정수 $a$, $b$($| a |, | b | \geq 3$)가 모든 양의 정수 $n$에 대해 $$P(n) \mid Q(n)^2 +aQ(n)+1, \quad Q(n) \mid P(n)^2+bP(n)+1$$ 을 만족한다. 이때 가능한 모든 $(a+b, \deg P)$ 순서쌍을 구하여라. [/hide]

Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that $$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Consider the set $A_n=\{x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n\}$ of $2n$ variables. How many permutations of set $A_n$ are there for which it is possible to assign real values from the interval $(0,1)$ to the $2n$ variables so that: (i) $x_i+y_i=1$ for each $i$; (ii) $x_1<x_2<\ldots<x_n$; (iii) the $2n$ terms of the permutation form a strictly increasing sequence?

Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same tick mark. If it is currently morning, how many minutes past midnight is it? [i]Ray Li[/i]

We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.

A fisherman, who was sailing in a rowing boat against the current of the river, had a hat falling from the bow of the boat into the water. After half an hour, the fisherman noticed the loss of his cap and immediately turned back. Find the speed of the river if the fisherman caught up with the cap at a distance of $a$ km from the place where it fell into the water (the speed of the river and the movement of the boat relative to the water is considered constant).

Rational number $\frac{m}{n}$ admits representation $$\frac{m}{n} = 1+ \frac12+\frac13 + ...+ \frac{1}{p-1}$$ where p $(p > 2)$ is a prime number. Show that the number $m$ is divisible by $p$.

Let $P,Q,R$ be points on the same side of a line $g$ in the plane. Let $M$ and $N$ be the feet of the perpendiculars from $P$ and $Q$ to $g$ respectively. Point $S$ lies between the lines $PM$ and $QN$ and satisfies and satisfies $PM = PS$ and $QN = QS$. The perpendicular bisectors of $SM$ and $SN$ meet in a point $R$. If the line $RS$ intersects the circumcircle of triangle $PQR$ again at $T$, prove that $S$ is the midpoint of $RT$.

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ The bisectors of the angles $B$ and $C$ meet each other in $I$ and meet the sides $AC$ and $AB$ in $D$ and $E$, respectively. Prove that $S_{BCDE}=2S_{BIC},$ where $S$ is the area function. [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttqqcc = rgb(0.2,0,0.8); pen qqwuqq = rgb(0,0.39,0); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2); pen ccccff = rgb(0.8,0.8,1); draw((1.89,4.08)--(1.89,4.55)--(1.42,4.55)--(1.42,4.08)--cycle,qqwuqq); draw((1.42,4.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(1.42,4.08),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(4,4.09),fftttt+linewidth(1.2pt)); draw((7.42,4.1)--(1.41,6.24),fftttt+linewidth(1.2pt)); draw((1.41,6.24)--(4,4.09),ccccff+linetype("5pt 5pt")); dot((1.42,4.08),ds); label("$A$", (1.1,3.66),NE*lsf); dot((7.42,4.1),ds); label("$B$", (7.15,3.75),NE*lsf); dot((1.4,10.08),ds); label("$C$", (1.49,10.22),NE*lsf); dot((4,4.09),ds); label("$E$", (3.96,3.46),NE*lsf); dot((1.41,6.24),ds); label("$D$", (0.9,6.17),NE*lsf); dot((3.37,5.54),ds); label("$I$", (3.45,5.69),NE*lsf); clip((-6.47,-7.49)--(-6.47,11.47)--(16.06,11.47)--(16.06,-7.49)--cycle); [/asy]

$a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$

Let $AA_1$ be an altitude in $\Delta ABC$. Let $H_a$ be the orthocenter of the triangle with vertices the tangential points of the excircle to $\Delta ABC$, opposite to $A$. The points $B_1$, $C_1$, $H_b$, and $H_c$ are defined analogously. Prove that $A_1 H_a$, $B_1 H_b$, and $C_1 H_c$ are concurrent.

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

Find all positive integers $m$ and $n$ such that $1 + 5 \cdot 2^m = n^2$.

Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils and $216$ erasers were distributed. This distribution was made so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. In this case, how many notebooks, pencils and erasers did each family receive?

Show that we cannot find $171$ binary sequences (sequences of $0$’s and $1$’s), each of length $12$ such that any two of them differ in at least four positions.

Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.

$A,B,C$ are playing backgammon tournament. At first, $A$ plays with $B$. Then the winner plays with $C$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that $C$ wins the tournament? $ \textbf{(A)}\ \frac27 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac3{14} \qquad\textbf{(D)}\ \frac 17 \qquad\textbf{(E)}\ \text{None} $

In an $N \times N$ table consisting of small unit squares, some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows, the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $N$?

Let $S$ be a set of $n$ points $P_1,P_2,\ldots,P_n$ in a plane such that no three of the points are collinear. Let $\alpha$ be the smallest of the angles $\angle P_iP_jP_k$ ($i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}$). Find $\max_S\alpha$ and determine those sets $S$ for which this maximal value is attained.

Let $ABC$ be an acute triangle with $AB < AC$, and let $H$ be its orthocenter. Let $D$, $E$, $F$ and $M$ be the midpoints of $BC$, $AC$, and $AH$, respectively. Prove that the circumcircles of triangles $AHD$, $BMC$, and $DEF$ pass through a common point.

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and $F_n = $ the remainder when $F_{n-1} + F_{n-2}$ is divided by $3$, for all $n \ge 2$. Thus the sequence starts $0,1,1,2,0,2 \ldots$. What is $F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]