Let $ABC$ be a triangle with $AB=5$, $AC=12$ and incenter $I$. Let $P$ be the intersection of $AI$ and $BC$. Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$, respectively, with centers $O_B$ and $O_C$. If the reflection of $BC$ over $AI$ intersects $\omega_B$ and $\omega_C$ at $X$ and $Y$, respectively, then $\frac{O_BO_C}{XY}=\frac{PI}{IA}$. Compute $BC$. [i]2016 CCA Math Bonanza Individual #15[/i]

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

Find $p+r$ if $p$ and $q$ are primes and $r$ is an integer such that \[ \left( r^2 + pr + 1 \right) \cdot \left( r^2 + \left( p^2 - q \right) r - p \right) = pq. \]

Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.

In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon. [i]Greece[/i]

Find all functions $f:(0,\infty)\to(0,\infty)$ such that $$f(f(f(x)))+4f(f(x))+f(x)=6x.$$

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

Bayus has eight slips of paper, which are labeled 1$, 2, 4, 8, 16, 32, 64,$ and $128.$ Uniformly at random, he draws three slips with replacement; suppose the three slips he draws are labeled $a, b,$ and $c.$ What is the probability that Bayus can form a quadratic polynomial with coefficients $a, b,$ and $c,$ in some order, with $2$ distinct real roots?

Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.

On a $9\times 9$ square lake composed of unit squares, there is a $2\times 4$ rectangular iceberg also composed of unit squares (it could be in either orientation; that is, it could be $4\times 2$ as well). The sides of the iceberg are parallel to the sides of the lake. Also, the iceberg is invisible. Lily is trying to sink the iceberg by firing missiles through the lake. Each missile fires through a row or column, destroying anything that lies in its row or column. In particular, if Lily hits the iceberg with any missile, she succeeds. Lily has bought $n$ missiles and will fire all $n$ of them at once. Let $N$ be the smallest possible value of $n$ such that Lily can guarantee that she hits the iceberg. Let $M$ be the number of ways for Lily to fire $N$ missiles and guarantee that she hits the iceberg. Compute $100M+N$. [i]Proposed by Brandon Wang[/i]

On the segment $ AC $ of the triangle $ ABC, $ let $ M $ be the midpoint of it, and let $ N $ a point on $ AM, $ distinct from $ A $ and $ M. $ The parallel through $ N $ with respect to $ AB $ intersects $ BM $ on $ P, $ the parallel through $ M $ with respect to $ BC $ intersects $ BN $ on $ Q, $ and the parallel through $ N $ with respect to $ AQ $ intersects $ BC $ on $ S. $ Prove that $ PS $ and $ AC $ are parallel.

Prove that every positive integer $k$ can be written as $k=\frac{mn+1}{m+n}$, where $m,n$ are positive integers.

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $ADX$ and $BCY$ meet line $XY$ again at $P$ and $Q$, respectively. Show that $OP=OQ$. [i]Holden Mui[/i]

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.

$ABCD$ is a square and $XYZ$ is an equilateral triangle such that $X$ lies on $AB$, $Y$ lies on $BC$ and $Z$ lies on $DA$. Line through the centers of $ABCD$ and $XYZ$ intersects $CD$ at $T$. Find angle $CTY$

Along the direct highway Tmutarakan - Uryupinsk at points $ A_1 $, $ A_2 $, $ \dots $, $ A_ {100} $ are the towers of the DPS mobile operator, and in points $ B_1 $, $ B_2 $, $ \dots $, $ B_ {100} $ are the towers of the "Horn" company. (Tower numbering may not coincide with the order of their location along the highway.) Each tower operates at a distance of $10$ km in both directions along the highway. It is known that $ A_iA_k \geq B_iB_k $ for any $ i $, $ k \leq 100 $. Prove that the total length of all sections of the highway covered by the DPS network is not less than the length of the sections covered by the Horn network .

Show that for any arbitrary triangle $ABC$, we have \[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]

A board $ n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? [asy] filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black); draw((4,0)--(4,5)); draw((3,0)--(3,5)); draw((2,0)--(2,5)); draw((1,0)--(1,5)); draw((0,4)--(5,4)); draw((0,3)--(5,3)); draw((0,2)--(5,2)); draw((0,1)--(5,1));[/asy] $ \textbf{(A)}\ 8:17\qquad\textbf{(B)}\ 25:49\qquad\textbf{(C)}\ 36:25\qquad\textbf{(D)}\ 32:17\qquad\textbf{(E)}\ 36:17 $

If $a,b,c>0$ then $$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$

Consider n real numbers $x_0, x_1, . . . , x_{n-1}$ for an integer $n \ge 2$. Moreover, suppose that for any integer $i$, $x_{i+n} = x_i$ . Prove that $$\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) \ge 0.$$