In $\triangle ABC$, $AB = 4$, $BC = 5$, and $CA = 6$. Circular arcs $p$, $q$, $r$ of measure $60^\circ$ are drawn from $A$ to $B$, from $A$ to $C$, and from $B$ to $C$, respectively, so that $p$, $q$ lie completely outside $\triangle ABC$ but $r$ does not. Let $X$, $Y$, $Z$ be the midpoints of $p$, $q$, $r$, respectively. If $\sin \angle XZY = \dfrac{a\sqrt{b}+c}{d}$, where $a, b, c, d$ are positive integers, $\gcd(a,c,d)=1$, and $b$ is not divisible by the square of a prime, compute $a+b+c+d$. [i]Proposed by Michael Tang[/i]

Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

Define a size-$n$ tromino to be the shape you get when you remove one quadrant from a $2n \times 2n$ square. In the figure below, a size-$1$ tromino is on the left and a size-$2$ tromino is on the right. [center][img]http://i.imgur.com/2065v7Y.png[/img][/center] We say that a shape can be tiled with size-$1$ trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-$1$ trominos. For example, a $23$ rectangle can be tiled with size-$1$ trominos as shown below, but a $33$ square cannot be tiled with size-$1$ trominos. [center][img]http://i.imgur.com/UBPeeRw.png[/img][/center] a) Can a size-$5$ tromino be tiled by size-$1$ trominos? b) Can a size-$2013$ tromino be tiled by size-$1$ trominos? Justify your answers.

An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$ B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.

(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that $$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$ (b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that $p(2018) = p(2019).$ Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$.

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?

Alice and Bob are builders; Charlie is a destroyer. Alice can build a car in $20$ hours and Bob can build a car in $10$ hours, while Charlie destroys a car in $40$ hours. If Alice and Bob are working together on a car Charlie is destroying, how many hours will it take for Alice and Bob to finish building the car?

Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? $\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$

For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? $ \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 20 \qquad \mathrm {(C) \ } 39 \qquad \mathrm{(D) \ } 49 \qquad \mathrm{(E) \ } 59$

Suppose that $p(n)$ is the number of ways to express $n$ as a sum of some naturall numbers (the two representations $4=1+1+2$ and $4=1+2+1$ are considered the same). Prove that for an infinite number of $n$'s $p(n)$ is even and for an infinite number of $n$'s $p(n)$ is odd.

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.

Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$. Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.

Prove that there exists an absolute constant $c$ such that any set $H$ of $n$ points of the plane in general position can be coloured with $c\log n$ colours in such a way that any disk of the plane containing at least one point of $H$ intersects some colour class of $H$ in exactly one point.

Let $n$ and $m$ be positive integers in the range $[1, 10^{10}]$. Let $R$ be the rectangle with corners at $(0, 0), (n, 0), (n, m), (0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called [i]far-reaching[/i] if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^6$.

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$. Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic. [i]Michael Ren[/i]

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$. [list=a] [*]Prove that $\triangle ALE\sim\triangle AEB$. [*]Prove that $FK=FL$. [/list]

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{ABD}$, if $AC<AD$ and $BC<BD$, then $\angle{C}>\angle{D}$. 35 points

Find all polynomials $f, g, h$ with real coefficients, such that $f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2$