For each pair of positive integers $(x, y)$ a nonnegative integer $x\Delta y$ is defined. It is known that for all positive integers $a$ and $b$ the following equalities hold: i. $(a + b)\Delta b = a\Delta b + 1$. ii. $(a\Delta b) \cdot (b\Delta a) = 0$. Find the values of the expressions $2016\Delta 121$ and $2016\Delta 144$.

Suppose that $x$ and $y$ are different real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for some four consecutive positive integers $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers n.

Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$

Let $D$ be a point different from the vertices on the side $BC$ of a triangle $ABC.$ Let $I, \: I_1$ and $I_2$ be the incenters of the triangles $ABC, \: ABD$ and $ADC,$ respectively. Let $E$ be the second intersection point of the circumcircles of the triangles $AI_1I$ and $ADI_2,$ and $F$ be the second intersection point of the circumcircles of the triangles $AII_2$ and $AI_1D.$ Prove that if $AI_1=AI_2,$ then \[ \frac{EI}{FI} \cdot \frac{ED}{FD}=\frac{{EI_1}^2}{{FI_1}^2}.\]

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$ [i]Proposed by Muztaba Syed[/i]

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Let $A$ be a set of $n\ge3$ points in the plane, no three of which are collinear. Show that there is a set $S$ of $2n-5$ points in the plane such that, for each triangle with vertices in $A$, there exists a point in $S$ which is strictly inside that triangle.

Find all functions $f\colon\mathbb{R}\to\mathbb{R}$ such that $(x-y)\bigl(f(x)+f(y)\bigr)\leqslant f\bigl(x^2-y^2\bigr)$ for all $x,y\in\mathbb{R}$.

There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition. [list] [*] [b](Condition)[/b] For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$. [/list]

Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.

For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.

Prove that $(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)$ if $a + b + c = 0$.

In triangle $ABC$, $\angle BAC=60^o$/ Let $\omega$ be a circle tangent to segment $AB$ at point $D$ and segment $AC$ at point $E$. Suppose $\omega$ intersects segment $BC$ at points $F$ and $G$ such that$ F$ lies in between $B$ and $G$. Given that $AD = FG = 4$ and $BF = \frac12$ , find the length of $CG$.

Prove that for every positive integer $n \geq 2$, $$\frac{\sum_{1\leq i \leq n} \sqrt[3]{\frac{i}{n+1}}}{n} \leq \frac{\sum_{1\leq i \leq n-1} \sqrt[3]{\frac{i}{n}}}{n-1}.$$

Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a=bc\sin^2A$. If $M$ is the maximum possible area of $ABC$ and $m$ is the minimum possible area of $ABC$, then $M^2+m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a+b$. [i]2016 CCA Math Bonanza Lightning #4.3[/i]

Let $O$ and $H$ be the circumcenter and orthocenter of a scalene triangle $ABC$, respectively. Let $D$ be the intersection point of the lines $AH$ and $BC$. Suppose the line $OH$ meets the side $BC$ at $X$. Let $P$ and $Q$ be the second intersection points of the circumcircles of $\triangle BDH$ and $\triangle CDH$ with the circumcircle of $\triangle ABC$, respectively. Show that the four points $P, D, Q$ and $X$ lie on a circle.

Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.

Find the next number in the sequence $131, 111311, 311321, 1321131211,\cdots$

In triangle $ ABC$, $ CD$ is the altitude to $ AB$ and $ AE$ is the altitude to $ BC.$ If the lengths of $ AB, CD,$ and $ AE$ are known, the length of $ DB$ is: $ \textbf{(A)}\ \text{not determined by the information given} \qquad$ $ \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad$ $ \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad$ $ \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad$ $ \textbf{(E)}\ \text{none of these is correct}$

Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.

Alfonso teaches Francis how to draw a spiral in the plane: First draw half of a unit circle. Starting at one of the ends, draw half a circle with radius $1/2$. Repeat this process at the endpoint of each half circle, where each time the radius is half of the previous half-circle. Assuming you can’t stop Francis from drawing the entire spiral, compute the total length of the spiral.

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

We are given a triangle $ ABC$ such that $ AC \equal{} BC$. There is a point $ D$ lying on the segment $ AB$, and $ AD < DB$. The point $ E$ is symmetrical to $ A$ with respect to $ CD$. Prove that: \[\frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}\]