Let $P_n$ denote the $n$-th Pell number defined by $P_{n+1}=2P_n+P_{n-1}$, $P_0=0$, $P_1=1$. Furthermore, let $T_n$ denote the $n$-th triangular number, that is $T_n=\binom{n+1}2$. Show that $$\sum_{n=0}^\infty4T_n\cdot\frac{P_n}{3^{n+2}}=P_3+P_4$$ [i]Proposed by Ángel Plaza[/i]

Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that \[ \angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ. \] Prove that \( \angle CDP = 30^\circ \). [i]Proposed by Vadym Solomka[/i]

The negation of the statement "No slow learners attend this school" is: $ \textbf{(A)}\ \text{All slow learners attend this school} \\ \textbf{(B)}\ \text{All slow learners do not attend this school} \\ \textbf{(C)}\ \text{Some slow learners attend this school} \\ \textbf{(D)}\ \text{Some slow learners do not attend this school} \\ \textbf{(E)}\ \text{No slow learners do not attend this school}$

Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

Two points are chosen uniformly at random on the sides of a square with side length 1. If $p$ is the probability that the distance between them is greater than $1$, what is $\lfloor 100p \rfloor$? (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. Let point $C$ be such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ at points $X$ and $Y$, respectively. Prove that $CX = CY$. [i]Proposed by Oleksii Masalitin[/i]

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. [i]Proposed by Merlijn Staps[/i]

Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$. [i]Proposed by Sharvil Kesarwani[/i]

At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines. a) Can there be an odd number of red lines if in the plane given there are no parallel lines? b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot BC$. Prove that there exists a triangle with edges $x, y, z.$

Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

Given a regular $72$-gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the $72$-gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.

There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation? [i]Author: Ray Li[/i]

a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$. b) for $n=3$ the same problem.

Let $P$ be an interior point of a triangle $ABC$ whose sidelengths are 26, 65, 78. The line through $P$ parallel to $BC$ meets $AB$ in $K$ and $AC$ in $L$. The line through $P$ parallel to $CA$ meets $BC$ in $M$ and $BA$ in $N$. The line through $P$ parallel to $AB$ meets $CA$ in $S$ and $CB$ in $T$. If $KL,MN,ST$ are of equal lengths, find this common length.

Let $P$ be a variable point on the circumference of a quarter-circle with radii $OA, OB$ and $\angle AOB = 90^\circ$. H is the projection of $P$ on $OA$. Find the locus of the incenter of the right-angled triangle $HPO.$

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane. Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano? [i]Proposed by Matthew Weiss

Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.

Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be points on $BC$ and $DC$ such that $\angle{EAF}=30^\circ$ and $CE=CF$. Determine the length of $BD$. [i]2015 CCA Math Bonanza Lightning Round #4.2[/i]

Prove that for infinitely many pairs $(a,b)$ of integers the equation \[x^{2012} = ax + b\] has among its solutions two distinct real numbers whose product is 1.