Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$

Let $a,b,c \in [1, 3]$ and satisfy the following conditions: $ max \{a, b, c\}\ge 2$ and $ a + b + c = 5$ What is the smallest possible value of $a^2 + b^2 + c^2$?

Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.

Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$. [i]Ray Li.[/i]

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$

There are two [i]allowed operations[/i] on a pair $(a, b)$ of positive integers: [list=i] [*]Add $1$ to both $a$ and $b$. [*]If one of the numbers $a$ or $b$ is a perfect cube, replace it with its cube root. [/list] The goal is to make the two numbers equal. Find all initial pairs $(a, b)$ for which this is possible.

The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

The figure below shows a $9\times7$ arrangement of $2\times2$ squares. Alternate squares of the grid are split into two triangles with one of the triangles shaded. Find the area of the shaded region. [asy] size(5cm); defaultpen(linewidth(.6)); fill((0,1)--(1,1)--(1,0)--cycle^^(0,3)--(1,3)--(1,2)--cycle^^(1,2)--(2,2)--(2,1)--cycle^^(2,1)--(3,1)--(3,0)--cycle,rgb(.76,.76,.76)); fill((0,5)--(1,5)--(1,4)--cycle^^(1,4)--(2,4)--(2,3)--cycle^^(2,3)--(3,3)--(3,2)--cycle^^(3,2)--(4,2)--(4,1)--cycle^^(4,1)--(5,1)--(5,0)--cycle,rgb(.76,.76,.76)); fill((0,7)--(1,7)--(1,6)--cycle^^(1,6)--(2,6)--(2,5)--cycle^^(2,5)--(3,5)--(3,4)--cycle^^(3,4)--(4,4)--(4,3)--cycle^^(4,3)--(5,3)--(5,2)--cycle^^(5,2)--(6,2)--(6,1)--cycle^^(6,1)--(7,1)--(7,0)--cycle,rgb(.76,.76,.76)); fill((2,7)--(3,7)--(3,6)--cycle^^(3,6)--(4,6)--(4,5)--cycle^^(4,5)--(5,5)--(5,4)--cycle^^(5,4)--(6,4)--(6,3)--cycle^^(6,3)--(7,3)--(7,2)--cycle^^(7,2)--(8,2)--(8,1)--cycle^^(8,1)--(9,1)--(9,0)--cycle,rgb(.76,.76,.76)); fill((4,7)--(5,7)--(5,6)--cycle^^(5,6)--(6,6)--(6,5)--cycle^^(6,5)--(7,5)--(7,4)--cycle^^(7,4)--(8,4)--(8,3)--cycle^^(8,3)--(9,3)--(9,2)--cycle,rgb(.76,.76,.76)); fill((6,7)--(7,7)--(7,6)--cycle^^(7,6)--(8,6)--(8,5)--cycle^^(8,5)--(9,5)--(9,4)--cycle,rgb(.76,.76,.76)); fill((8,7)--(9,7)--(9,6)--cycle,rgb(.76,.76,.76)); draw((0,0)--(0,7)^^(1,0)--(1,7)^^(2,0)--(2,7)^^(3,0)--(3,7)^^(4,0)--(4,7)^^(5,0)--(5,7)^^(6,0)--(6,7)^^(7,0)--(7,7)^^(8,0)--(8,7)^^(9,0)--(9,7)); draw((0,0)--(9,0)^^(0,1)--(9,1)^^(0,2)--(9,2)^^(0,3)--(9,3)^^(0,4)--(9,4)^^(0,5)--(9,5)^^(0,6)--(9,6)^^(0,7)--(9,7)); draw((0,1)--(1,0)^^(0,3)--(3,0)^^(0,5)--(5,0)^^(0,7)--(7,0)^^(2,7)--(9,0)^^(4,7)--(9,2)^^(6,7)--(9,4)^^(8,7)--(9,6)); [/asy]

Two squares on an $8\times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.

Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

For a nonempty set $A$ of integers, let $\mathrm{range} \, A=\max A-\min A$. Find the number of subsets $S$ of \[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] such that $\mathrm{range} \, S$ is an element of $S$.

For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?

Positive integers $x,y,z$ satisfy $x^3+xy+x^2+xz+y+z=301$. Compute $y+z-x$. [i]2015 CCA Math Bonanza Individual Round #12[/i]

Using digits $ 1, 2, 3, 4, 5, 6$, without repetition, $ 3$ two-digit numbers are formed. The numbers are then added together. Through this procedure, how many different sums may be obtained?

Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.

Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$