Let $A$, $B$ and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a$, $b$, $c$ be the distances from $P$ to $A$, $B$, $C$ respectively. Show that there is a triangle with side lengths $a$, $b$, $c$, and that the area of this triangle depends only on the distance from $P$ to $O$.

Which of the numbers $2^{2002}$ and $2000^{200}$ is bigger?

The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$, where $m$, $n$, and $p$ are integers. Find $m+n+p$.

Is it possible to place $1965$ points in a square with side $1$ so that any rectangle of area $1/200$ with sides parallel to the sides of the square contains at least one of these points inside?

Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.

The $120$ permutations of the word BORIS are arranged in alphabetical order, from BIORS to SROIB. What is the $60$th word in this list?

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ -1\qquad\textbf{(E)}\ -19 $

Let $\alpha, c > 0$, define $x_1 = c$ and let $x_{n + 1} = x_n e^{-x_n^\alpha}$ for $n \geq 1$. For which values of $\beta$ does $\sum_{i = 1}^{\infty} x_n^\beta$ converge?

Source: Lusophon MO 2016 Prove that any positive power of $2$ can be written as: $$5xy-x^2-2y^2$$ where $x$ and $y$ are odd numbers.

Consider a semicircle with center $M$ and diameter $AB$. Let $P$ be a point in the semicircle, different from $A$ and $B$, and let $Q$ be the midpoint of the arc $AP$. The line parallel to $QP$ through $M$ intersects $PB$ at the point $S$. Prove that the triangle $PMS$ is isosceles.

A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to $\mathcal{P}.$ [center] [img] https://cdn.artofproblemsolving.com/attachments/a/8/7607ee5d199471fd13b09a41a473c71d5d935b.png [/img] [/center]

A tangent $l$ to the circle inscribed in a rhombus meets its sides $AB$ and $BC$ at points $E$ and $F$ respectively. Prove that the product $AE\cdot CF$ is independent of the choice of $l$.

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?

Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$.

Suppose the real numbers $a, A, b, B$ satisfy the inequalities: $$|A - 3a| \le 1 - a\,\,\, , \,\,\, |B -3b| \le 1 - b$$, and $a, b$ are positive. Prove that $$\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.$$

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

If $1$ in every $20$ people is left handed, what is the expected number of left handed people in a group of $400$ people? $\textbf{(A) } 0.05\qquad\textbf{(B) } 5\qquad\textbf{(C) } 15\qquad\textbf{(D) } 20\qquad\textbf{(E) } 200$

Let $M,N,P$ be points in the sides $BC,AC,AB$ of $\triangle ABC$ respectively. The quadrilateral $MCNP$ has an incircle of radius $r$, if the incircles of $\triangle BPM$ and $\triangle ANP$ also have the radius $r$. Prove that $$AP\cdot MP=BP\cdot NP$$

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

Annie has a permutation $(a_1, a_2, \dots ,a_{2019})$ of $S=\{1,2,\dots,2019\}$, and Yannick wants to guess her permutation. With each guess Yannick gives Annie an $n$-tuple $(y_1, y_2, \dots, y_{2019})$ of integers in $S$, and then Annie gives the number of indices $i\in S$ such that $a_i=y_i$. (a) Show that Yannick can always guess Annie's permutation with at most $1200000$ guesses. (b) Show that Yannick can always guess Annie's permutation with at most $24000$ guesses. [i]Yannick Yao[/i]

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

From a sequence of integers $(a, b, c, d)$ each of the sequences \[(c, d, a, b),\quad (b, a, d, c),\quad (a + nc, b + nd, c, d),\quad (a + nb, b, c + nd, d)\] for arbitrary integer $n$ can be obtained by one step. Is it possible to obtain $(3, 4, 5, 7)$ from $(1, 2, 3, 4)$ through a sequence of such steps?

$I,\Omega$ are the incenter and the circumcircle of triangle $ABC$, respectively, and the tangents of $B,C$ to $\Omega$ intersect at $L$. Assume that $P\neq C$ is a point on $\Omega$ such that $CI,AP$, and the circle with center $L$ and radius $LC$ are concurrent. Let the foot from $I$ to $AB$ be $F$, the midpoint of $BC$ be $M$, $X$ is a point on $\Omega$ s.t. $AI,BC,PX$ are concurrent. Prove that the lines $AI,AX,MF$ form an isosceles triangle. [i]Proposed by ckliao914[/i]

Let $ABCD$ be a cyclic quadrilateral, with circumcircle $\omega$ and circumcenter $O$. Let $AB$ intersect $CD$ at $E$, $AD$ intersect $BC$ at $F$, and $AC$ intersect $BD$ at $G$. The points $A_1, B_1, C_1, D_1$ are chosen on rays $GA$, $GB$, $GC$, $GD$ such that: $\bullet$ $\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}$ $\bullet$ The points $A_1, B_1, C_1, D_1, O$ lie on a circle. Let $A_1B_1$ intersect $C_1D_1$ at $K$, and $A_1D_1$ intersect $B_1C_1$ at $L$. Prove that the image of the circle $(A_1B_1C_1D_1)$ under inversion about $\omega$ is a line passing through the midpoints of $KE$ and $LF$. [i]Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin[/i]