Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$, \[a^n + a^{n-1} +
\cdots
+ a^1 + 1 \mid a^{n!} + a^{(n-1)!} +
\cdots

+ a^{1!} + 1.\] [i]Proposed by Milos Milosavljevic[/i]

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$ (b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.

Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.

There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$. After all 1011 people have gone by, how many light bulbs are on?

Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$, $$ f(xy-x)+f(x+f(y))=yf(x)+3 $$

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n\geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p$. Find $p$.

Albert writes down all of the multiples of $9$ between $9$ and $999,$ inclusive. Compute the sum of the digits he wrote.

Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that: [list] [*]The sum of the fractions is equal to $2$. [*]The sum of the numerators of the fractions is equal to $1000$. [/list] In how many ways can Pedro do this?

Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that $$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$

Suppose you are given $n$ blocks, each of which weighs an integral number of pounds, but less than $n$ pounds. Suppose also that the total weight of the $n$ blocks is less than $2n$ pounds. Prove that the blocks can be divided into two groups, one of which weighs exactly $n$ pounds.

Ari the Archer is shooting at an abnormal target. The target consists of $100$ concentric rings, each of width $1$, so that the total radius of the target is $100$. The point value of a given ring of the target is equal to its area (so getting a bull's eye would be worth $\pi$ points, but hitting on the outer ring would give $199\pi$ points). Given that Ari hits any point on the target uniformly at random, what is his expected score? [i]2017 CCA Math Bonanza Individual Round #7[/i]

2004 computers make up a network using several cables. If for a subset $S$ in the set of all computers, there isn't a cable that connects two computers in $S$, $S$ is called independant. One lets the arbitrary independant set consists at most 50 computers, and uses the least number of cables. (1) Let $c(L)$ be the number of cables which connects the computer $L$. Prove that for two computers $A,B$, $c(A)=c(B)$ if there is a cable which connects $A$ and $B$, $|c(A)-c(B)|\leq 1$ otherwise. (2) Determine the number of used cables.

Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area

The sales tax rate in Bergville is $6\%$. During a sale at the Bergville Coat Closet, the price of a coat is discounted $20\%$ from its \$90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up \$90.00 and adds $6\%$ sales tax, then subtracts $20\%$ from this total. Jill rings up \$90.00, subtracts $20\%$ of the price, then adds $6\%$ of the discounted price for sales tax. What is Jack's total minus Jill's total? $ \textbf{(A)}\ -\$1.06\qquad\textbf{(B)}\ -\$0.53\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \$0.53\qquad\textbf{(E)}\ \$1.06 $

Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

How many triangles appear in the diagram below: [asy] import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.6)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=0,xmax=8,ymin=0,ymax=8; draw((0,8)--(0,0)); draw((0,0)--(8,0)); draw((8,0)--(8,8)); draw((8,8)--(0,8)); draw((0,8)--(1,7)); draw((1,7)--(2,8)); draw((2,8)--(3,7)); draw((3,7)--(4,8)); draw((4,8)--(5,7)); draw((5,7)--(6,8)); draw((6,8)--(7,7)); draw((7,7)--(8,8)); draw((8,6)--(7,7)); draw((0,6)--(1,7)); draw((1,7)--(2,6)); draw((2,6)--(3,7)); draw((3,7)--(4,6)); draw((4,6)--(5,7)); draw((5,7)--(6,6)); draw((6,6)--(7,7)); draw((1,5)--(0,6)); draw((1,5)--(2,6)); draw((2,6)--(3,5)); draw((3,5)--(4,6)); draw((4,6)--(5,5)); draw((5,5)--(6,6)); draw((6,6)--(7,5)); draw((7,5)--(8,6)); draw((7,5)--(8,4)); draw((0,4)--(1,5)); draw((1,5)--(2,4)); draw((2,4)--(3,5)); draw((3,5)--(4,4)); draw((4,4)--(5,5)); draw((5,5)--(6,4)); draw((6,4)--(7,5)); draw((1,3)--(0,4)); draw((1,3)--(2,4)); draw((3,3)--(4,4)); draw((3,3)--(2,4)); draw((5,3)--(4,4)); draw((5,3)--(6,4)); draw((6,4)--(7,3)); draw((7,3)--(8,4)); draw((8,2)--(7,3)); draw((0,2)--(1,3)); draw((1,3)--(2,2)); draw((2,2)--(3,3)); draw((3,3)--(4,2)); draw((5,3)--(4,2)); draw((5,3)--(6,2)); draw((7,3)--(6,2)); draw((7,1)--(6,2)); draw((7,1)--(8,2)); draw((7,1)--(8,0)); draw((6,0)--(7,1)); draw((4,0)--(5,1)); draw((5,1)--(6,0)); draw((2,0)--(3,1)); draw((3,1)--(4,0)); draw((0,0)--(1,1)); draw((1,1)--(2,0)); draw((1,1)--(0,2)); draw((1,1)--(2,2)); draw((2,2)--(3,1)); draw((3,1)--(4,2)); draw((4,2)--(5,1)); draw((5,1)--(6,2)); dot((8,0),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

Let $a,b,c$ be real numbers satisfying $a<b<c,a+b+c=6,ab+bc+ac=9$. Prove that $0<a<1<b<3<c<4$ [hide="Solution"] Let $abc=k$, then $a,b,c\ (a<b<c)$ are the roots of cubic equation $x^3-6x^2+9x-k=0\Longleftrightarrow x(x-3)^2=k$ that is to say, $a,b,c\ (a<b<c)$ are the $x$-coordinates of the interception of points between $y=x(x-3)^2$ and $y=k$. $y=x(x-3)^2$ have local maximuml value of $4$ at $x=1$ and local minimum value of $0$ at $x=3$. Since the $x$-coordinate of the interception point between $y=x(x-3)^2$ and $y=4$ which is the tangent line at local maximum point $(1,4)$ is a point $(4,4)$,Moving the line $y=k$ so that the two graphs $y=x(x-3)^2$ and $y=k$ have the distinct three interception points,we can find that the range of $a,b,c$ are $0<a<1,1<b<3,3<c<4 $,we are done.[/hide]

Let $(x_1,x_2,\dots,x_{100})$ be a permutation of $(1,2,...,100)$. Define $$S = \{m \mid m\text{ is the median of }\{x_i, x_{i+1}, x_{i+2}\}\text{ for some }i\}.$$ Determine the minimum possible value of the sum of all elements of $S$.

A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? $\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$

let ${a_{n}}$ be a sequence of integers,$a_{1}$ is odd,and for any positive integer $n$,we have $n(a_{n+1}-a_{n}+3)=a_{n+1}+a_{n}+3$,in addition,we have $2010$ divides $a_{2009}$ find the smallest $n\ge\ 2$,so that $2010$ divides $a_{n}$

For $ s,t \in \mathbb{N}$, consider the set $ M\equal{}\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$. Find the number of rhombi with the vertices in $ M$ and the diagonals parallel to the coordinate axes.