Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.

What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$? $ \textbf{(A)}\ 972 \qquad\textbf{(B)}\ 971 \qquad\textbf{(C)}\ 970 \qquad\textbf{(D)}\ 969 \qquad\textbf{(E)}\ 968 $

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list? [i]2016 CCA Math Bonanza Individual #3[/i]

Evin’s calculator is broken and can only perform $3$ operations: Operation $1$: Given a number $x$, output $2x$. Operation $2$: Given a number $x$, output $4x +1$. Operation $3$: Given a number $x$, output $8x +3$. After initially given the number $0$, how many numbers at most $128$ can he make?

Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and each cevian starting from that vertex, is possible to construct a triangle.

$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$. Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.

Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made $ 48$ free throws. How many free throws did she make at the first practice? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$

A sequence of numbers $ a_1, a_2, a_3, ...$ satisfies (i) $ a_1 \equal{} \frac{1}{2}$ (ii) $ a_1\plus{}a_2 \plus{} \cdots \plus{} a_n \equal{} n^2 a_n \ (n \geq 1)$ Determine the value of $ a_n \ (n \geq 1)$.

Show that $e=\sum^{\infty}_{n=0} \frac{1}{n!}$ is irrational.

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

Find all values of the positive integer $k$ that has the property: There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.

For each integer $n \ge 3$ solve in real numbers the system of equations: $$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$

Prove that for positive real numbers $a, b$ and $c$ the inequality $2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2$ holds if and only if $a,b,c$ are the sides of a triangle.

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

What is the remainder when $2^{1023}$ is divided by $1023$?

In trapezoid $ABCD$ holds $AD \mid \mid BC$, $\angle ABC = 30^{\circ}$, $\angle BCD = 60^{\circ}$ and $BC=7$. Let $E$, $M$, $F$ and $N$ be midpoints of sides $AB$, $BC$, $CD$ and $DA$, respectively. If $MN=3$, find $EF$

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.

Indicate the moment in time when for the first time after midnight the angle between the minute and hour hands will be equal to $1^o$, despite the fact that the minute hand shows an integer number of minutes.

In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]"Arc $BC$ of the circumcircle" means "the arc with endpoints $B$ and $C$ not containing $A$".[/list][/hide]

Given is the equation $(m, n) +[m, n] =m+n$ where $m, n$ are positive integers and m>n. a) Prove that n divides m. b) If $m-n=10$, solve the equation.