Let $XY Z$ be a triangle with $\angle XY Z = 40^o$ and $\angle Y ZX = 60^o$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $XY Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{XI}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle AIB$.

Compute $$\prod^{12}_{k=1} \left(\prod^{10}_{j=1} \left(e^{2\pi ji/11} - e^{2\pi ki/13}\right) \right) .$$ (The notation $\prod^{b}_{k=a}f(k)$means the product $f(a)f(a + 1)... f(b)$.)

Let $x_1,x_2,...,x_k$ be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a [b]scramble[/b] if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is $1,3,3,5$ then $3,5,1,3$ is a scramble, but $3,3,1,5$ is not. A rearrangement is called a [b]two-two[/b] if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3,5,1,3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$). Let $n\geq 2$. Prove that the number of scrambles of $1,1,2,3,...,n-1,n$ is equal to the number of two-twos of $1,2,3,...,n,n+1$. (Notice that both sequences have $n+1$ numbers, but the first one contains two 1s.)

Determine the maximum possible value of $$\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})$$ over all $x\in [0,1]$.

Define a sequence $(a_n)$ by $a_1 =1, a_2 =2,$ and $a_{k+2}=2a_{k+1}+a_k$ for all positive integers $k$. Determine all real numbers $\beta >0$ which satisfy the following conditions: (A) There are infinitely pairs of positive integers $(p,q)$ such that $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$ (B) There are only finitely many pairs of positive integers $(p,q)$ with $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$ for which there is no index $k$ with $q=a_k.$

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6 $

Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and \[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$. Prove that they are converges and find their limits.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.

Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.

There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?

Let be a real number $ a\in (0,1) $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that: $$ \lim_{x\to 0} f(x) =0= \lim_{x\to 0} \frac{f(x)-f(ax)}{x} $$ Prove that $ \lim_{x\to\infty } \frac{f(x)}{x} =0. $

Circumferences $C_1$ and $C_2$ have different radios and are externally tangent on point $T$. Consider points $A$ on $C_1$ and $B$ on $C_2$, both different from $T$, such that $\angle BTA=90^{\circ}$. What is the locus of the midpoints of line segments $AB$ constructed that way?

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

Divide a cube into three equal pyramids.

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

Jenny is going to attend a sports camp for $7$ days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of $4$ consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.

How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $

A plane geometric figure of $n$ sides with the vertices $A_1,A_2,A_3,\dots, A_n$ ($A_i$ is adjacent to $A_{i+1}$ for every $i$ integer where $1\leq i\leq n-1$ and $A_n$ is adjacent to $A_1$) is called [i]brazilian[/i] if: I - The segment $A_jA_{j+1}$ is equal to $(\sqrt{2})^{j-1}$, for every $j$ with $1\leq j\leq n-1$. II- The angles $\angle A_kA_{k+1}A_{k+2}=135^{\circ}$, for every $k$ with $1\leq k\leq n-2$. [b]Note 1:[/b] The figure can be convex or not convex, and your sides can be crossed. [b]Note 2:[/b] The angles are in counterclockwise. a) Find the length of the segment $A_nA_1$ for a brazilian figure with $n=5$. b) Find the length of the segment $A_nA_1$ for a brazilian figure with $n\equiv 1$ (mod $4$).

Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms. [i]Proposed by Oleg Mushkarov and Nikolai Nikolov[/i]

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

Let $P(x)$ be a polynomial such that $P(x^2 - 1) = x^4 - 3x^2 + 3$. Find $P(x^2 + 1)$.

Find the remainder when $A=3^3\cdot 33^{33}\cdot 333^{333}\cdot 3333^{3333}$ is divided by $100$.

For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows: a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle A_1B_1C_1$ consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on. Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.