A dot is marked at each vertex of a triangle $ABC$. Then, $2$, $3$, and $7$ more dots are marked on the sides $AB$, $BC$, and $CA$, respectively. How many triangles have their vertices at these dots?

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of $\Delta ABC$, and $NP$, $NT$ be the tangents to the incircle of this triangle. The lines $BP$ and $BT$ meet the circumcircle for the second time at points $P_1$ and $T_1$ respectively. Prove that $PP_1 = TT_1$.

How many integer solutions are there to $y^2=x^2-2017$?

Evaluate \[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\] [i]Proposed by Nathan Xiong[/i]

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? [i]Proposed by Espen Slettnes[/i]

Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andy Xu[/i]

Show that there does not exist a right triangle with all integer side lengths such that exactly one of the side lengths is odd.

Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$

$$Problem 1 $$The set S = {1,2,3,...,2006} is partitioned into two disjoint subsets A and B such that: (i) 13 ∈ A; (ii) if a ∈ A, b ∈ B, a+b ∈ S, then a+b ∈ B; (iii) if a ∈ A, b ∈ B, ab ∈ S, then ab ∈ A. Determine the number of elements of A

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively. Suppose that $\frac{ax + b}{x} \in Q$ for every $x \in N^*$: Prove that there exist integers $A,B,C$ such that $\frac{ax + b}{x}= \frac{Ax + B}{Cx}$ for all $x \in N^* $

In the acute triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to $BC$, let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be a point on the line segment $DE$. Prove that $AF$ is perpendicular to $BE$ if and only if $FE/FD = BD/DC$

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

Prove that if $ n $ is a natural number and the angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then $$\frac{1}{\sin 2\alpha} + \frac{1}{\sin 4\alpha} + \frac{1}{\sin 8\alpha} + ... + = ctg \alpha - ctg 2^n \alpha.$$