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]

Evaluate : \[ \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} \] Express your expression in the form $\frac{a+b\sqrt{c}}{d}$ where $a,b,c,d\in \Bbb{Z}$.

Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i},$ where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers, and let $f(n) = \prod_{i=1}^k e_i^{p_i}.$ Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128.$

A finite set of positive integers $A$ is called [i]meanly[/i] if for each of its nonempy subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if $\frac{1}{k}(a_1 + \dots + a_k)$ is an integer whenever $k \ge 1$ and $a_1, \dots, a_k \in A$ are distinct. Given a positive integer $n$, determine the least possible sum of the elements of a meanly $n$-element set.

Beatriz has three dice on whose faces different letters are written. By rolling all three dice on one table, and choosing each time only the letters of the faces above, she formed the words $$OSA , VIA , OCA , ESA , SOL , GOL , FIA , REY , SUR , MIA , PIO , ATE , FIN , VID.$$ Determine the six letters of each die.

Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+4)$ is an integer. (Author: O. Podlipski)

Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.