Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.

Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.

At a tourist camp, each person has at least $50$ and at most $100$ friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) $1331$ different colors, and any person has $20$ friends whose t-shirts all have pairwisely different colors.

Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$.

In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$. [img]https://cdn.artofproblemsolving.com/attachments/2/9/0884eb311d1e40300c1e5980fd53eaadfa7a25.png[/img]

For a non-empty finite complex number set $A$, define the "[i]Tao root[/i]" of $A$ as $\left|\sum_{z\in A} z \right|$. Given the integer $n\ge 3$, let the set $$U_n = \{\cos\frac{2k \pi}{n}+ i\sin\frac{2k \pi}{n}|k=0,1,...,n-1\}.$$Let $a_n$ be the number of non-empty subsets in which the [i]Tao root [/i] of $U_n$ is $0$ , $b_n$ is the number of non-empty subsets of $U_n$ whose [i]Tao root[/i] is $1$. Compare the sizes of $na_n$ and $2b_n$.

Find the sum of all positive integers $n$ with the following properties: [list] [*] $n$ is not divisible by any primes larger than $10$. [*] For some positive integer $k$, the positive divisors of $n$ are \[1=d_1<d_2<d_3\cdots<d_{2k}=n.\] [*] The divisors of $n$ have the property that \[d_1+d_2+\cdots+d_k=3k.\] [/list] [i]2021 CCA Math Bonanza Team Round #7[/i]

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.

[asy] size(150); pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25); draw(circle(A,1)^^D--B); draw(circle(E,.5)^^circle( F ,.25)); label("$A$", D, W); label("$K$", A, S); label("$B$", B, dir(0)); label("$L$", E, N); label("$M$",shift(-.05,.05)*F); //Credit to Klaus-Anton for the diagram[/asy] In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of circle $\mathit{K}$ to the area of circle $\mathit{M}$ is $\textbf{(A) }12\qquad\textbf{(B) }14\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad \textbf{(E) }\text{not an integer}$

Compute the number of tuples $(a_0, a_1, a_2, a_3, a_4, a_5)$ of (not necessarily positive) integers such that $a_i \le i$ for all $0 \le i \le 5$ and $$a_0+a_1+a_2+a_3+a_4+a_5=6.$$

Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

Let $M$ be the midpoint of side $BC$ of triangle $ABC$, $Q$ the point of intersection of its angle bisectors. It is known that $MQ=QA$. Find the smallest possible value of angle $\angle MQA$.

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.

In a company of 7 sousliks each souslik has 4 friends. Is it always possible to find in this company two non-intersecting groups of 3 sousliks each such that in both groups all sousliks are friends?

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to [asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75)); [/asy] $ \textbf{(A)}\ k \qquad\textbf{(B)}\ \frac{6k}{5} \qquad\textbf{(C)}\ \frac{9k}{8} \qquad\textbf{(D)}\ \frac{5k}{4} \qquad\textbf{(E)}\ 2k $

A crymble is a solid consisting of four white and one black unit cubes as shown in the picture. Find the side length of the smallest cube that can be exactly filled up with crymbles. [img]https://cdn.artofproblemsolving.com/attachments/b/0/b1e50f7abbfb7d356913d746d653fd3875f5ae.png[/img]

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

$2025$ light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $k$ light bulbs. [i]Based on an OKTV problem[/i]

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]