Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$. Find $$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}$$

Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon. [i]Proposed by Hesam Rajabzade[/i]

A positive integer $n$ is called [i]beautiful[/i] if, for every integer $4 \le b \le 10000$, the base-$b$ representation of $n$ contains the consecutive digits $2$, $0$, $2$, $3$ (in this order, from left to right). Determine whether the set of all beautiful integers is finite. [i]Oleg Kryzhanovsky[/i]

In the country of [i]Sugarland[/i], there are $13$ students in the IMO team selection camp. $6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all $13$ students have a chance of being a team member? [i]Proposed by Morteza Saghafian[/i]

Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that: (i) $f(1)=1$, (ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$, (iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.

A number line with the integers $1$ through $20,$ from left to right, is drawn. Ten coins are placed along this number line, with one coin at each odd number on the line. A legal move consists of moving one coin from its current position to a position of strictly greater value on the number line that is not already occupied by another coin. How many ways can we perform two legal moves in sequence, starting from the initial position of the coins (different two-move sequences that result in the same position are considered distinct)?

Consider three sequences $(a_n)_{n=1}^{^\infty}$, $(b_n)_{n=1}^{^\infty}$ , $(c_n)_{n=1}^{^\infty}$, each of which has pairwisedistinct terms. Prove that there exist two indices $k$ and $l$ for which $k < l$, $$a_k < a_l , b_k < b_l , \,\,\, and \,\,\, c_k < c_l.$$

Let $Y$ be the family of all subsets of $X = \{1,2,...,n\}$ ($n > 1$) and let $f : Y \to X$ be an arbitrary mapping. Prove that there exist distinct subsets $A,B$ of $X$ such that $f(A) = f(B) = max A\triangle B$, where $A\triangle B = (A-B)\cup(B-A)$.

Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$. $F$ is a point on $\Gamma_1$, the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$. $MG$ is a line tangent to $\Gamma_2$ at $G$. Prove that $MF=MG$.

A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$

A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.

The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.

Clara computed the product of the first $n$ positive integers, and Valerie computed the product of the first $m$ even positive integers, where $m\ge2$. They got the same answer. Prove that one of them had made a mistake.

Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.

Hello, I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated. Compute the following primitive: \[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]

A positive integer $n$ is called $\textit{un-two}$ if there does not exist an ordered triple of integers $(a,b,c)$ such that exactly two of$$\dfrac{7a+b}{n},\;\dfrac{7b+c}{n},\;\dfrac{7c+a}{n}$$are integers. Find the sum of all un-two positive integers. [i]Proposed by [b]stayhomedomath[/b][/i]

Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone? $\textbf{(A) }1:1 \qquad \textbf{(B) }47:43 \qquad \textbf{(C) }2:1 \qquad \textbf{(D) }40:13 \qquad \textbf{(E) }4:1$ [asy] size(350); defaultpen(linewidth(0.8)); real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q; path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9); draw(ellipse(origin,r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill(origin--Pp--Qp--cycle,gray(0.8)); draw((-r,h1)--(0,0)--(r,h1)^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(0,Qp.y),Arrows(size=8)); draw(origin--(0,12),linetype("4 4")); draw(origin--(r*(s-0.1),0)); label("$3$",(-0.9,h1*s),N,fontsize(10)); real h2 = 7.5, r = 6, s=0.6, d = 14; pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0); path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1); draw(ellipse((d,0),r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill((d,0)--Pp--Qp--cycle,gray(0.8)); draw(P--(d,0)--Q^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype("4 4")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(d,Qp.y),Arrows(size=8)); draw((d,0)--(d,10),linetype("4 4")); draw((d,0)--(d+r*(s-0.1),0)); label("$6$",(d-r/4,h2*s-0.06),N,fontsize(10)); [/asy]

[b]1.[/b] Given a positive integer $r>1$, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers $S_{1},S_{2},\dots,S_{r}$ such that $S_{1}+S_{2}+\dots+S_{r}=1$, any of these infinite geometrical series can be divided into $r$ infinite series(not necessarily geometrical) having the sums $S_{1},S_{2},\dots,S_{r}$, respectively. [b](S. 6)[/b]

Let $k,n\in\mathbb{N},0<k<n.$ Prove that \[\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.\]

$ R_k(m,n)$ is the least number such that for each coloring of $ k$-subsets of $ \{1,2,\dots,R_k(m,n)\}$ with blue and red colors, there is a subset with $ m$ elements such that all of its k-subsets are red or there is a subset with $ n$ elements such that all of its $ k$-subsets are blue. a) If we give a direction randomly to all edges of a graph $ K_n$ then what is the probability that the resultant graph does not have directed triangles? b) Prove that there exists a $ c$ such that $ R_3(4,n)\geq2^{cn}$.

Positive reals $a$, $b$, and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$. Prove that \[ \sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}. \][i]Proposed by Evan Chen[/i]

The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$.

Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals $\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$? $\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$