Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled. Liana Topan

A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?

A positive integer $n$ is given. If there exist sets $F_1, F_2, \cdots F_m$ satisfying the following, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements in $A$. $A-B$ is the set of elements that are in $A$ but not $B$) (i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots n\}$ (ii): $|F_1| \le |F_2| \le \cdots \le |F_m|$ (iii): For all $1 \le i < j \le m$, $|F_i-F_j|=1$.

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$, then the following conditions hold: $\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$; $\bullet$ the number $Q(0)$ is divisible by $n$; and $\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$? Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$. [b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar. [b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$. [b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$. [b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$. PS. You had better use hide for answers.

Solve in integers the equation $xy + 3x - 5y = - 3$.

Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$ [i]Proposed by Michael Tang[/i]

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

Let $ p_1,p_2,\dots,p_r$ be distinct primes, and let $ n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $ (x, y)$ such that \[ x^3 \plus{} y^3 \equal{} p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}\] does not exceed $ 2^{r\plus{}1}$.

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{more than }6$

$f(x)=ax^2+bx+c;a,b,c$ are reals. $M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\}$ Prove that $M=N$ or $M \cap N = \O $

Echo the gecko starts on the point $(0, 0)$ in the 2D coordinate plane. Every minute, starting at the end of the first minute, he'll teleport $1$ unit up, left, right, or down with equal probability. Echo dies the moment he lands on a point that is more than $1$ unit away from the origin. The average number of minutes he'll live can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Team #2[/i]

For exactly two real values of $b$, $b_1$ and $b_2$, the line $y=bx-17$ intersects the parabola $y=x^2 +2x+3$ at exactly one point. Compute $b_1^2+b_2^2$.

Let $AA_1$ and $BB_1$ be heights in acute triangle intersects at $H$. Let $A_1A_2$ and $B_1B_2$ be heights in triangles $HBA_1$ and $HB_1A$, respe. Prove that $A_2B_2$ and $AB$ are parralel.

Four people are playing rock-paper-scissors. They each play one of the three options (rock, paper, or scissors) independently at random, with equal probability of each choice. Compute the probability that someone beats everyone else. (In rock-paper-scissors, a player that plays rock beats a player that plays scissors, a player that plays paper beats a player that plays rock, and a player that plays scissors beats a player that plays paper.)

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]

Find all natural numbers $ n,x,y $ such that $ \big| 2^x-n^{y+1}\big| =1 . $

Every vertex of a convex polyhedron belongs to three edges. It is possible to circumscribe a circle around all its faces. Prove that the polyhedron can be inscribed in a sphere.