A group of $101$ Dalmathians participate in an election, where they each vote independently on either candidate $A$ or $B$ with equal probability. If $X$ Dalmathians voted for the winning candidate, the expected value of $X^2$ can be expressed as $\tfrac{a}{b}$ for positive integers $a,b$ with $\gcd(a,b) = 1.$ Find the unique positive integer $k \le 103$ such that $103 | a-bk.$

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$) goes through a narrow slit which has width $d$. After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $. Find the width of the slit $d$, in nanometers. [i](Proposed by Ahaan Rungta)[/i]

The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

Find all positive integers $m$ such that $$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$

In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$. [i]Proposed by Tejaswi Navilarekallu[/i]

Let $S$ be a finite set of points in the plane,situated in general position(any three points in $S$ are not collinear),and let $$D(S,r)=\{\{x,y\}:x,y\in S,\text{dist}(x,y)=r\},$$ where $R$ is a positive real number,and $\text{dist}(x,y)$ is the euclidean distance between points $x$ and $y$.Prove that $$\sum_{r>0}|D(S,r)|^2\le\frac{3|S|^2(|S|-1)}{4}.$$

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$? $\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.

Find all positive integers $x , y$ which are roots of the equation $2 x^y-y= 2005$ [u] Babis[/u]

Let $x$ and $y$ be positive real numbers such that $y^3 + y \leq x - x^3$. Prove that (A) $y < x < 1$ (B) $x^2 + y^2 < 1$.

Let $a,b,c$ be real numbers for which $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximal value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$.

For integer $n\geq 2$ and real $0\leq p\leq 1$, define $\mathcal{W}_{n,p}$ to be the complete weighted undirected random graph with vertex set $\{1,2,\ldots,n\}$: the edge $(i,j)$ will have weight $\min(i,j)$ with probability $p$ and weight $\max(i,j)$ otherwise. Let $\mathcal{L}_{n,p}$ denote the total weight of the minimum spanning tree of $\mathcal{W}_{n,p}$. Find the largest integer less than the expected value of $\mathcal{L}_{2018,1/2}$.

Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$. Remark: "Natural numbers" is the set of positive integers.

Let $ABCD$ is a trapezoid with $\angle A = \angle B = 90^o$ and let $E$ is a point lying on side $CD$. Let the circle $\omega$ is inscribed to triangle $ABE$ and tangents sides $AB, AE$ and $BE$ at points $P, F$ and $K$ respectively. Let $KF$ intersects segments $BC$ and $AD$ at points $M$ and $N$ respectively, as well as $PM$ and $PN$ intersect $\omega$ at points $H$ and $T$ respectively. Prove that $PH = PT$.

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 10 $

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

Let $ABC$ be an acute-angled triangle, with $\angle B \neq \angle C$ . Let $M$ be the midpoint of side $BC$, and $E,F$ be the feet of the altitude from $B,C$ respectively. Denote by $K,L$ the midpoints of segments $ME,MF$, respectively. Suppose $T$ is a point on the line $KL$ such that $AT//BC$. Prove that $TA=TM$ .

Two integers are called [i]relatively prime[/i] if they share no common factors other than $1.$ Determine the sum of all positive integers less than $162$ that are relatively prime to $162.$

[b]p5.[/b] Find the number of three-digit integers that contain at least one $0$ or $5$. The leading digit of the three-digit integer cannot be zero. [b]p6.[/b] What is the sum of the solutions to $\frac{x+8}{5x+7} =\frac{x+8}{7x+5}$ [b]p7.[/b] Let $BC$ be a diameter of a circle with center $O$ and radius $4$. Point $A$ is on the circle such that $\angle AOB = 45^o$. Point $D$ is on the circle such that line segment$ OD$ intersects line segment $AC$ at $E$ and $OD$ bisects $\angle AOC$. Compute the area of $ADE$, which is enclosed by line segments $AE$ and $ED$ and minor arc $AD$. [b]p8. [/b] William is a bacteria farmer. He would like to give his fiance$ 2021$ bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those $2021$ bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals? PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].