Let $n$ be an integer greater than $1$. In how many ways can we fill all the numbers $1, 2,..., 2n$ in the boxes of a grid of $2\times n$, one in each box, so that any two consecutive numbers are they in squares that share one side of the grid?

Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

Let $ABCDEF$ be an inscribed hexagon with $$AB.CD.EF=BC.DE.FA$$ Let $B_1$ be the reflection point of $B$ with respect to $AC$ and $D_1$ be the reflection point of $D$ with respect to $CE,$ and finally let $F_1$ be the reflection point of $F$ with respect to $AE.$ Prove that $\triangle B_1D_1F_1\sim BDF.$

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

The altitudes of an acute triangle $ABC$ intersect at $H$. The tangent line at $H$ to the circumcircle of triangle $BHC$ intersects the lines $AB$ and $AC$ at points $Q$ and $P$ respectively. The circumcircles of triangles $ABC$ and $APQ$ intersect at point $K$ ($K\neq A$). The tangent lines at the points $A$ and $K$ to the circumcircle of triangle $APQ$ intersect at $T$. Prove that $TH$ passes through the midpoint of segment $BC$.

Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color. [b] proposed by S. Berlov[/b]

A rabbit is placed on a $2n\times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares. (Adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit form a polygon $\mathcal{P}$. Find the maximum possible number of the vertices of $\mathcal{P}$. For example the answer for the case $n=2$ would be $12$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(2cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -11.3, xmax = 27.16, ymin = -11.99, ymax = 10.79; /* image dimensions */ /* draw figures */ draw((5.14,3.19)--(8.43,3.22), linewidth(1)); draw((8.43,3.22)--(11.72,3.25), linewidth(1)); draw((11.72,3.25)--(11.75,-0.04), linewidth(1)); draw((11.75,-0.04)--(11.78,-3.33), linewidth(1)); draw((11.78,-3.33)--(8.49,-3.36), linewidth(1)); draw((8.49,-3.36)--(5.2,-3.39), linewidth(1)); draw((5.2,-3.39)--(5.17,-0.1), linewidth(1)); draw((5.17,-0.1)--(5.14,3.19), linewidth(1)); draw((6.785,3.205)--(6.845,-3.375), linewidth(1)); draw((8.43,3.22)--(8.49,-3.36), linewidth(1)); draw((10.075,3.235)--(10.135,-3.345), linewidth(1)); draw((5.155,1.545)--(11.735,1.605), linewidth(1)); draw((5.17,-0.1)--(11.75,-0.04), linewidth(1)); draw((11.765,-1.685)--(5.185,-1.745), linewidth(1)); draw((5.97,2.375)--(10.905,2.42), linewidth(1)); draw((10.905,2.42)--(10.92,0.775), linewidth(1)); draw((10.92,0.775)--(9.275,0.76), linewidth(1)); draw((9.275,0.76)--(9.29,-0.885), linewidth(1)); draw((9.29,-0.885)--(10.935,-0.87), linewidth(1)); draw((10.935,-0.87)--(10.95,-2.515), linewidth(1)); draw((10.95,-2.515)--(6.015,-2.56), linewidth(1)); draw((6.015,-2.56)--(6,-0.915), linewidth(1)); draw((6,-0.915)--(7.645,-0.9), linewidth(1)); draw((7.645,-0.9)--(7.63,0.745), linewidth(1)); draw((7.63,0.745)--(5.985,0.73), linewidth(1)); draw((5.985,0.73)--(5.97,2.375), linewidth(1)); /* dots and labels */ dot((5.97,2.375),linewidth(4pt) + dotstyle); dot((5.985,0.73),linewidth(4pt) + dotstyle); dot((6,-0.915),linewidth(4pt) + dotstyle); dot((6.015,-2.56),linewidth(4pt) + dotstyle); dot((7.645,-0.9),linewidth(4pt) + dotstyle); dot((7.63,0.745),linewidth(4pt) + dotstyle); dot((9.275,0.76),linewidth(4pt) + dotstyle); dot((9.29,-0.885),linewidth(4pt) + dotstyle); dot((10.95,-2.515),linewidth(4pt) + dotstyle); dot((10.935,-0.87),linewidth(4pt) + dotstyle); dot((10.92,0.775),linewidth(4pt) + dotstyle); dot((10.905,2.42),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts: [list] (a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$; (b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers. (c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds. [/list] Find all possible values of $v_s, v_p$ and $v_d$.

Let $S$ be the set of ordered integer pairs $(x, y)$ such that $0 < x < y < 42$ and there exists some integer $n$ such that $x^6-y^6 \mid n^2+2015^2$. What is the sum $\sum_{(x_i, y_i) \in S}x_iy_i$?

Two squids are forced to participate in a game. Before it begins, they will be informed of all the rules, and can discuss their strategies freely. Then, they will be locked in separate rooms, and be given distinct positive integers no larger than $2023$ as their IDs respectively. The two squids then take turns alternatively; on one's turn, the squid chooses one of the following: 1. announce a positive integer, which will be heard by the other squid; 2. declare which squid has the larger ID. If correct, they win and are released together; otherwise, they lose and are fried together. Find the smallest positive integer $N$ so that, no matter what IDs the squids have been given, they can always win in a finite number of turns, and the sum of the numbers announced during the game is no larger than $N$.

Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy: $\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$ What values can be taken by the product $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have? [i]2018 CCA Math Bonanza Lightning Round #4.2[/i]

For which integers $n \ge 5$ is it possible to color the vertices of a regular$ n$-gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?

Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$

Evaluate \[ \sqrt{\binom82+\binom92+\binom{15}2+\binom{16}2}. \] [i] Proposed by Evan Chen [/i]

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

Some coins are placed on a $20 \times 13$ board. Two coins are called [i]neighbours[/i] if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.

Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is $ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$

An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.

In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions $ \{(0,0),(0,1),(1,0),(1,1)\}$. Given a finite number of moves can you yield the configuration $ \{(0,0), (1,1), (3,0), (2, \minus{} 1)\}$ ?

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

Given a triangular pyramid $SABC$, in which $\angle BSC = \alpha$, $\angle CSA =\beta$, $\angle ASB = \gamma$, and the dihedral angles at the edges $SA$ and $SB$ have the value of $\phi$ and $\delta$, respectively. Prove that $\gamma > \alpha \cdot \cos \delta +\beta \cdot \cos \phi.$$

[b]6.[/b] Show that the number of the faces of a convex polyhedron is even if every face is centrally simmetric. [b](G. 12)[/b]