Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$. Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$. Time allowed for this problem was 1 hour.

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Let $\Omega$ be a circle centered at $O$. Let $ABCD$ be a quadrilateral inscribed in $\Omega$, such that $AB = 12$, $AD = 18$, and $AC$ is perpendicular to $BD$. The circumcircle of $AOC$ intersects ray $DB$ past $B$ at $P$. Given that $\angle PAD = 90^\circ$, find $BD^2$.

Let $n \geq 3$ be a positive integer. A [i]chipped $n$-board[/i] is a $2 \times n$ checkerboard with the bottom left square removed. Lino wants to tile a chipped $n$-board and is allowed to use the following types of tiles: [list] [*] Type 1: any $1 \times k$ board where $1 \leq k \leq n$ [*] Type 2: any chipped $k$-board where $1 \leq k \leq n$ that must cover the left-most tile of the $2 \times n$ checkerboard. [/list] Two tilings $T_1$ and $T_2$ are considered the same if there is a set of consecutive Type 1 tiles in both rows of $T_1$ that can be vertically swapped to obtain the tiling $T_2$. For example, the following three tilings of a chipped $7$-board are the same: [img]http://i.imgur.com/8QaSgc0.png[/img] For any positive integer $n$ and any positive integer $1 \leq m \leq 2n - 1$, let $c_{m,n}$ be the number of distinct tilings of a chipped $n$-board using exactly $m$ tiles (any combination of tile types may be used), and define the polynomial $$P_n(x) = \sum^{2n-1}_{m=1} c_{m,n}x^m.$$ Find, with justification, polynomials $f(x)$ and $g(x)$ such that $$P_n(x) = f(x)P_{n-1}(x) + g(x)P_{n-2}(x)$$ for all $n \geq 3$.

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine all positive integers \( K \) such that the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(2x + 3y + 3z) = K \] holds for some positive integers $x,y,z$.

Find the triplets of primes $(a,\ b,\ c)$ such that $a-b-8$ and $b-c-8$ are primes.

An ant is moving on the cooridnate plane, starting form point $(0,-1)$ along a straight line until it reaches the $x$- axis at point $(x,0)$ where $x$ is a real number. After it turns $90^o$ to the left and moves again along a straight line until it reaches the $y$-axis . Then it again turns left and moves along a straight line until it reaches the $x$-axis, where it once more turns left by $90^o$ and moves along a straight line until it finally reached the $y$-axis. Can both the length of the ant's journey and distance between it's initial and final point be: (a) rational numbers ? (b) integers? Justify your answers PS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here[/url]

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.

Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

What is the integer closest to $\pi^{\pi}$? (No calculator allowed!) [i]2015 CCA Math Bonanza Lightning Round #5.1[/i]

Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.

Let $ P $ be a point on the side $ AB $ of the triangle $ ABC. $ The parallels through $ P $ of the medians $ AA_1,BB_1 $ intersect $ BC,AC $ at $ R,Q, $ respectively. Show that $ P, $ the middlepoint of $ RQ $ and the centroid of $ ABC $ are collinear.

Let $n_1,n_2,\ldots,n_s$ be distinct integers such that $$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample: $(\text a)$ $|n_i|=1$ for some $i$ $(\text b)$ If further all $n_i$ are positive, then $$\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.$$

Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$. Note: The set of points that satisfy a property is called a locus.

Prove that the equation \[ x^z + y^z = z^z \] has no solutions in postive integers.

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

A frog is standing in a center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with their current square uniformly at random, and jumps onto the lilypad on their chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of times the frog jumps? [i]2021 CCA Math Bonanza Lightning Round #3.2[/i]

Construct outer squares $ABMN$, $BCKL$, $ACPQ$ on sides $[AB]$, $[BC]$, $[CA]$ of triangle $ABC$, respectively. Construct squares $NQZT$ and $KPYX$ on segments $[NQ]$ and $[KP]$. If $Area(ABMN) - Area(BCKL)=1$, what is $Area(NQZT)-Area(KPYX)$? $ \textbf{(A)}\ \dfrac 34 \qquad\textbf{(B)}\ \dfrac 53 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$

Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show that $x^3_1$ and $x^3_2$ are the roots of $$y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.$$

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$. [i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]