Let $XYZ$ be a triangle with $ \angle X = 60^\circ $ and $ \angle Y = 45^\circ $. A circle with center $P$ passes through points $A$ and $B$ on side $XY$, $C$ and $D$ on side $YZ$, and $E$ and $F$ on side $ZX$. Suppose $AB=CD=EF$. Find $ \angle XPY $ in degrees.

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $£$, lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$. The ray $PK$ meets $\omega$ at point $Q$. Prove that the circumcircle of triangle $EQF$ touches $\omega$.

Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following: \begin{align*} &(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\ &(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\ &(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\ &(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5}) \end{align*} Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.

Given two distinct numbers $ b_1$ and $ b_2$, their product can be formed in two ways: $ b_1 \times b_2$ and $ b_2 \times b_1.$ Given three distinct numbers, $ b_1, b_2, b_3,$ their product can be formed in twelve ways: $ b_1\times(b_2 \times b_3);$ $ (b_1 \times b_2) \times b_3;$ $ b_1 \times (b_3 \times b_2);$ $ (b_1 \times b_3) \times b_2;$ $ b_2 \times (b_1 \times b_3);$ $ (b_2 \times b_1) \times b_3;$ $ b_2 \times(b_3 \times b_1);$ $ (b_2 \times b_3)\times b_1;$ $ b_3 \times(b_1 \times b_2);$ $ (b_3 \times b_1)\times b_2;$ $ b_3 \times(b_2 \times b_1);$ $ (b_3 \times b_2) \times b_1.$ In how many ways can the product of $ n$ distinct letters be formed?

Find all pairs $(m, n)$ of integers for which $$\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.$$

Let $G$ be the set of $2\times 2$ matrices that such $$ G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid\, a,b,c,d \in \mathbb{Z}, ad-bc = 1, c \text{ is a multiple of } 3 \right\} $$ and two matrices in $G$: $$ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\;\;\; B = \begin{pmatrix} -1 & 1 \\ -3 & 2 \end{pmatrix} $$ Show that any matrix in $G$ can be written as a product $M_1M_2\cdots M_r$ such that $M_i \in \{A, A^{-1}, B, B^{-1}\}, \forall i \leq r$

Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.

Kermit writes down the numbers $1$, $2$, $3$, $4$, $5$. He then erases one number, and discovers that the sum of the remaining numbers is $13$. Which number was erased? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

A man walked a certain distance at a constant rate. If he had gone $\textstyle\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\textstyle\frac{1}{2}$ mile per hour slower, he would have been $2\textstyle\frac{1}{2}$ hours longer on the road. The distance in miles he walked was $\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }15\qquad\textbf{(C) }17\frac{1}{2}\qquad\textbf{(D) }20\qquad \textbf{(E) }25$

There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper. Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation). [i]Proposed by Stephan Wagner, Austria[/i]

The tangent lines from a point $P$ meet a circle $k$ at $A$ and $B$. Let $X$ be an arbitrary point on the shorter arc $AB$, and $C$ and $D$ be the orthogonal projections of $P$ onto the lines $AX$ and $BX$, respectively. Prove that the line $CD$ passes through a fixed point $Y$ as $X$ moves along the arc $AB$.

Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane. Note that the $x$ coodinate of $P$ is less than that of $Q$. (a) If the origin $O$ is the midpoint of the lines　egment $PQ$, then find the equation of the line $PQ$. (b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$. (c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.

Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than $1$ that divides every term of the sequence?

Let \(m,n\) and \(a\) be positive integers. Lumis has \(m\) cards, each with the number \(n\) written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than \(\displaystyle\frac{n}{2^m}\). Prove that if \(n\) is replaced everywhere by \(a\), then the resulting expression will have the same value as before or will be undefined due to division by zero.

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list] [*] $a_1 \geq 2018^{2018};$ [*] $a_m \leq a_n$ whenever $m \leq n$; [*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$? [/list] [i](Dominic Yeo, United Kingdom)[/i]

Let $A_1B_1C_1D_1E_1$ be a regular pentagon.For $ 2 \le n \le 11$, let $A_nB_nC_nD_nE_n$ be the pentagon whose vertices are the midpoint of the sides $A_{n-1}B_{n-1}C_{n-1}D_{n-1}E_{n-1}$. All the $5$ vertices of each of the $11$ pentagons are arbitrarily coloured red or blue. Prove that four points among these $55$ points have the same colour and form the vertices of a cyclic quadrilateral.

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that \[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\] for all $n \ge 1$.

Given a set $S$ consisting of two undefined elements "pib" and "maa", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Consider the three theorems: $T_1$: There are exactly six maas, $T_2$: There are exactly three maas in each pib, $T_3$: For each maa there is exactly one other maa not in the same pid with it. The theorems which are deducible from the postulates are: $\textbf{(A)}\ T_3 \; \text{only}\qquad \textbf{(B)}\ T_2 \; \text{and} \; T_3 \; \text{only} \qquad \textbf{(C)}\ T_1 \; \text{and} \; T_2 \; \text{only}\\ \textbf{(D)}\ T_1 \; \text{and} \; T_3 \; \text{only}\qquad \textbf{(E)}\ \text{all}$

Find the smallest positive integer $n$ such that there exists a prime $p$ where $p$ and $p + 10$ both divide $n$ and the sum of the digits of $n$ is $p$.

Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.

The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$. Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively . a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$. b) Denote $min \,\{AC, BD\} = m$. Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$.

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$: $$\cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right).$$ [i]2017 CCA Math Bonanza Individual Round #10[/i]

Let $ABC$ be a non-degenerate triangle inscribed in a circle, such that $AB$ is the diameter of the circle. Let the angle bisectors of the angles at $A$ and $B$ meet at $P.$ Determine the maximum possible value of $\angle APB,$ in degrees.