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 $x$.

Inside the convex polyhedron, the point $P$ and several lines $\ell_1,\ell_2, ..., \ell_n$ passing through $P$ and not lying in the same plane. To each face of the polyhedron we associate one of the lines $l_1, l_2, ..., l_n$ that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.

A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.

In the 27 points of a cube: 8 vertexes, 12 midpoints of edges, 6 centers of surfaces, and the center of the cube, the number of groups of three collinear points is $\text{(A)}57\qquad\text{(B)}49\qquad\text{(C)}43\qquad\text{(D)}37$

Mellon Game Lab has come up with a concept for a new game: Square Finder. The premise is as follows. You are given an $n\times n$ grid of squares (for integer $n\geq 2$), each of which is either blank or has an arrow pointing up, down, left, or right. You are also given a $2\times 2$ grid of squares that appears somewhere in this grid, possibly rotated. For example, see if you can find the following $2\times 2$ grid inside the larger $4\times 4$ grid. [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,u},{r,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] [asy] size(4cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{u,b,b,r},{b,r,u,d},{d,b,u,b},{u,r,b,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Did you spot it? It's in the bottom left, rotated by $90^\circ$ clockwise. To make the game as interesting as possible, Mellon Game Lab would like the grid to be as large as possible and for no $2\times 2$ grid to appear more than once in the big grid. The grid above doesn't work, as the following $2\times 2$ grid appears twice, once in the top left corner (rotated $90^\circ$ counterclockwise) and once directly below it (overlapping). [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,r},{d,b}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Let's call a grid that avoids such repeats a [i]repeat-free grid[/i]. We are interested in finding out for which $n$ constructing an $n\times n$ repeat-free grid is possible. Here's what we know so far. [list] [*] Any $2\times 2$ grid is repeat-free, as there is only one subgrid to worry about, and there can't possibly be any repeats. [*] If we can construct an $n\times n$ repeat-free grid, we can also construct a $k\times k$ repeat-free grid for any $k\leq n$ by just taking the top left $k\times k$ of the original one we found. [*] By the previous observation, if it is impossible to construct such an $n\times n$ repeat-free grid, we cannot construct a $k\times k$ repeat-free grid for any $k\geq n$, as otherwise we could take the top left $n\times n$ to get one working for $n$. [/list] These three observations together tell us that either we can construct an $n\times n$ repeat-free grid for all $n\geq 2$, or there exists some upper limit $N\geq 2$ such that we can construct an $n\times n$ repeat-free grid for all $n\leq N$ but cannot construct one for any $n> N$. Your goal is to determine if such an $N$ exists, and if so, place bounds on its value. More precisely, this problem consists of two parts: a lower bound and an upper bound. For the lower bound, to show that $N\geq n$ for some $n$, you need to construct an $n\times n$ repeat-free grid (you do not need to prove your construction works). For the upper bound, to show that $N$ is at most some value $n$, you must prove that it is impossible to construct an $(n+1)\times (n+1)$ repeat-free grid. [i]Proposed by Connor Gordon and Eric Oh[/i]

We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.

Jerry has a four-sided die, a six-sided die, and an eight-sided die. Each die is numbered starting at one. Jerry rolls the three dice simultaneously. What is the probability that they all show different numbers? $\mathrm{(A) \ } \frac{35}{48} \qquad \mathrm{(B) \ } \frac{35}{64} \qquad \mathrm {(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{5}{12} \qquad \mathrm{(E) \ } \frac{5}{8}$

[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $ [b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality $$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

Let $ABC$ be a triangle. Point $D$ lies on segment $BC$ such that $\angle BAD = \angle DAC$. Point $X$ lies on the opposite side of line $BC$ as $A$ and satisfies $XB=XD$ and $\angle BXD = \angle ACB$. The point $Y$ is defined similarly. Prove that the lines $XY$ and $AD$ are perpendicular.

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

What is the perimeter of trapezoid $ ABCD$? [asy]defaultpen(linewidth(0.8));size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label("30", (9,12), W); label("50", (43,24), N); label("25", (71.5, 12), E); label("24", (18, 12), E); label("$A$", a, SW); label("$B$", b, N); label("$C$", c, N); label("$D$", d, SE); label("$E$", e, S);[/asy] $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 $