Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds $$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$

Given a $2 \times 2$ tile and seven dominoes ( $2 \times 1$ tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a $2 \times 7$ rectangle using some of these tiles.

Between the states of Alinaesquina and Berlinda, each road connects one city of Alinaesquina to one city of Berlinda. All the roads are in two-ways, and between any two cities, it is possible to travel from one to the other, using only these (possibly more than one) roads. Furthermore, it is known that, from any city of anyone of the two states, the same number of $k$ roads are going out. We know that $k\geq 2$. Prove that governments of the states can close anyone of the roads, and there will still be a route (possibly through several roads) between any two cities.

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.

Prove that for all positive and admissible values of $x$ the following inequality holds: $$\sin x + arc \sin x>2x$$

If $\log_2(\log_{\frac{1}{2}}(\log_2x))=\log_3(\log_{\frac{1}{3}}(\log_3y))=\log_5(\log_{\frac{1}{5}}(\log_5z))=0$, then $\text{(A)}z<x<y\qquad\text{(B)}x<y<z\qquad\text{(C)}y<z<x\qquad\text{(D)}z<y<x$

The area of a convex quadrilateral is $S$, and the angle between the diagonals is $a$. On the sides of this quadrilateral, as on the bases, isosceles triangles with vertex angle equal to $\phi$, wherein two opposite triangles are located on the other side of the corresponding side of the quadrilateral than the quadrilateral itself, and the other two are located on the other side. Prove that the vertices of the constructed triangles, different from the vertices of the quadrilateral, serve as the vertices of a parallelogram. Find the area of this parallelogram.

$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right|\equal{}\left|BE\right|\equal{}\sqrt{2}$, find the area of triangle $ AFC$. $\textbf{(A)}\ \frac{7\sqrt{3} }{2} \qquad\textbf{(B)}\ \frac{9\sqrt{2} }{2} \qquad\textbf{(C)}\ 4\sqrt{2} \qquad\textbf{(D)}\ 2\sqrt{3} \qquad\textbf{(E)}\ 2\sqrt{2}$

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

A circle is randomly chosen in a circle of radius $1$ in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?

Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.

Consider a natural number $n\geqslant 3$ and a graph $G{}$ with a chromatic number $\chi(G)=n$ which has more than $n{}$ vertices. Prove that there exist two vertex-disjoint subgraphs $G_1{}$ and $G_2{}$ of $G{}$ such that $\chi(G_1)+\chi(G_2)\geqslant n+1.$ [i]Proposed by V. Dolnikov[/i]

A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

Alice, Beth, Carla, Dana, and Eden play a game in which each of them simultaneously closes her eyes and randomly chooses two of the others to point at (one with each hand). A participant loses if she points at someone who points back at her; otherwise, she wins. Find the probability that all five girls win.

Let $a, b,$ and $c$ be non-zero real numbers and let $a \ge b \ge c$. Prove the inequality $\frac{a^3 - c^3}{3} \ge abc (\frac{a- b}{c}+ \frac{b- c}{a})$ . When does equality hold?

Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

Triangle $ABC$ has points $A$ at $\left(0,0\right)$, $B$ at $\left(9,12\right)$, and $C$ at $\left(-6,8\right)$ in the coordinate plane. Find the length of the angle bisector of $\angle{BAC}$ from $A$ to where it intersects $BC$. [i]2017 CCA Math Bonanza Lightning Round #1.3[/i]

Fill in the grid below with the numbers $1$ through $25$, with each number used exactly once, subject to the following constraints: [list=1] [*] Each shaded square contains an even number, and each unshaded square contains an odd number. [/*] [*] For any pair of squares that share a side, if $x$ and $y$ are the two numbers in those squares, then either $x\geq2y$ or $y\geq2x$. [/*] [/list] Four numbers have been filled in already. [asy] size(10cm); for(int i=1; i<5; ++i){ draw((-2i+1,-2i+9)--(2i-1,-2i+9)); draw((-2i+1,2i-9)--(2i-1,2i-9)); draw((-2i+9,-2i+1)--(-2i+9,2i-1)); draw((2i-9,-2i+1)--(2i-9,2i-1)); } for(int i=1; i<3; ++i){ filldraw((-1,2i+1)--(-1,2i-1)--(-3,2i-1)--(-3,2i+1)--cycle,lightgray); } for(int i=2; i<4; ++i){ filldraw((1,2i+1)--(1,2i-1)--(-1,2i-1)--(-1,2i+1)--cycle,lightgray); } for(int i=1; i<5; ++i){ filldraw((3,2i-5)--(3,2i-7)--(1,2i-7)--(1,2i-5)--cycle,lightgray); } filldraw((-5,1)--(-5,-1)--(-7,-1)--(-7,1)--cycle,lightgray); filldraw((-1,-1)--(-1,-3)--(-3,-3)--(-3,-1)--cycle,lightgray); filldraw((1,-5)--(1,-7)--(-1,-7)--(-1,-5)--cycle,lightgray); filldraw((5,3)--(5,1)--(3,1)--(3,3)--cycle,lightgray); label("\Huge{25}",(-4,2)); label("\Huge{13}",(0,0)); label("\Huge{16}",(0,6)); label("\Huge{21}",(4,-2)); [/asy]

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Prove that, if every three consecutive vertices of a convex $n{}$-gon, $n\geqslant 4$, span a triangle of area at least 1, then the area of the $n{}$-gon is (strictly) greater than $(n\log_2 n)/4-1/2.$ [i]Radu Bumbăcea & Călin Popescu[/i]

[u]Round 1[/u] [b]p1.[/b] Ten children arrive at a birthday party and leave their shoes by the door. All the children have different shoe sizes. Later, as they leave one at a time, each child randomly grabs a pair of shoes their size or larger. After some kids have left, all of the remaining shoes are too small for any of the remaining children. What is the greatest number of shoes that might remain by the door? [b]p2.[/b] Turans, the king of Saturn, invented a new language for his people. The alphabet has only $6$ letters: A, N, R, S, T, U; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the first word is SATURN. Which word follows immediately after TURANS? [b]p3.[/b] Benji chooses five integers. For each pair of these numbers, he writes down the pair's sum. Can all ten sums end with different digits? [b]p4.[/b] Nine dwarves live in a house with nine rooms arranged in a $3\times3$ square. On Monday morning, each dwarf rubs noses with the dwarves in the adjacent rooms that share a wall. On Monday night, all the dwarves switch rooms. On Tuesday morning, they again rub noses with their adjacent neighbors. On Tuesday night, they move again. On Wednesday morning, they rub noses for the last time. Show that there are still two dwarves who haven't rubbed noses with one another. [b]p5.[/b] Anna and Bobby take turns placing rooks in any empty square of a pyramid-shaped board with $100$ rows and $200$ columns. If a player places a rook in a square that can be attacked by a previously placed rook, he or she loses. Anna goes first. Can Bobby win no matter how well Anna plays? [img]https://cdn.artofproblemsolving.com/attachments/7/5/b253b655b6740b1e1310037da07a0df4dc9914.png[/img] [u]Round 2[/u] [b]p6.[/b] Some boys and girls, all of different ages, had a snowball fight. Each girl threw one snowball at every kid who was older than her. Each boy threw one snowball at every kid who was younger than him. Three friends were hit by the same number of snowballs, and everyone else took fewer hits than they did. Prove that at least one of the three is a girl. [b]p7.[/b] Last year, jugglers from around the world travelled to Jakarta to participate in the Jubilant Juggling Jamboree. The festival lasted $32$ days, with six solo performances scheduled each day. The organizers noticed that for any two days, there was exactly one juggler scheduled to perform on both days. No juggler performed more than once on a single day. Prove there was a juggler who performed every day. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Quadrilateral $ABCD$ is inscribed in a circle, $E$ is an arbitrary point of this circle. It is known that distances from point $E$ to lines $AB, AC, BD$ and $CD$ are equal to $a, b, c$ and $d$ respectively. Prove that $ad= bc$.

Let $P_1,P_2,\ldots ,P_n$, be infinite arithmetic progressions of positive integers, of differences $d_1,d_2,\ldots ,d_n$, respectively. Prove that if every positive integer appears in at least one of the $n$ progressions then one of the differences $d_i$ divides the least common multiple of the remaining $n-1$ differences. Note: $P_i=\left \{ a_i,a_i+d_i,a_i+2d_i,a_i+3d_i,a_i+4d_i,\cdots \right \}$ with $ a_i$ and $d_i$ positive integers.

Using one straight cut we partition a rectangular piece of paper into two pieces. We call this one "operation". Next, we cut one of the two pieces so obtained once again, to partition [i]this piece[/i] into two smaller pieces (i.e. we perform the operation on any [i]one[/i] of the pieces obtained). We continue this process, and so, after each operation we increase the number of pieces of paper by $1$. What is the minimum number of operations needed to get $47$ pieces of $46$-sided polygons? [obviously there will be other pieces too, but we will have at least $47$ (not necessarily [i]regular[/i]) $46$-gons.]

Determine the value of the sum \[\left|\sum_{1\leq i<j\leq 50}ij(-1)^{i+j}\right|.\]