Let $S = \{1, 2, 3, . . . , 64\}.$ Compute the number of ways to partition $S$ into $16$ arithmetic sequences such that each arithmetic sequence has length $4$ and common difference $1, 4,$ or $16.$

Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $ [i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $

When the circumference of a toy balloon is increased from $20$ inches to $25$ inches, the radius is increased by: $\textbf{(A)}\ 5\text{ in} \qquad \textbf{(B)}\ 2\dfrac{1}{2}\text{ in} \qquad \textbf{(C)}\ \dfrac{5}{\pi}\text{ in} \qquad \textbf{(D)}\ \dfrac{5}{2\pi}\text{ in} \qquad \textbf{(E)}\ \dfrac{\pi}{5}\text{ in}$

Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$

A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.

The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$, and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$. The segment $AD$ meets $\omega$ for the second time at point $Z$. Prove that the circumcircle of the triangle $XYZ$ touches $\omega$.

Let a triangle $\triangle ABC$ and the real numbers $x$, $y$, $z>0$. Prove that $$x^n\cos\frac{A}{2}+y^n\cos\frac{B}{2}+z^n\cos\frac{C}{2}\geq (yz)^{\frac{n}{2}}\sin A +(zx)^{\frac{n}{2}}\sin B +(xy)^{\frac{n}{2}}\sin C$$

[i]a)[/i] The solid $S$ is defined as the intersection of the six spheres with the six edges of a regular tetrahedron $T$, with edge length $1$, as diameters. Prove that $S$ contains two points at a distance $\frac{1}{\sqrt 6}.$ [i]b)[/i] Using the same assumptions in [i]a)[/i], prove that no pair of points in $S$ has a distance larger than $\frac{1}{\sqrt 6}.$

Two distinct positive factors of $144$ are selected at random. What is the probability that their product is greater than $144$?

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

Let $n \ge 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_1,a_2,\cdots, a_n$ be the lengths of sides of $P$ and let $p$ be its perimeter. Prove that \[\frac{a_1}{p-a_1}+\frac{a_2}{p-a_2}+\cdots + \frac{a_n}{p-a_n} < 2 \]

Given real numbers $a, b, c$ such that $a + b - c = ab- bc - ca = abc = 8$. Find all possible values of $a$.

The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities $$a+b < c+d < e+a < b+c < d+e.$$ Among the five numbers, which one is the smallest, and which one is the largest?

Jack and Jill run $10$ kilometers. They start at the same point, run $5$ kilometers up a hill, and return to the starting point by the same route. Jack has a $10$ minute head start and runs at the rate of $15$ km/hr uphill and $20$ km/hr downhill. Jill runs $16$ km/hr uphill and $22$ km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? $ \textbf{(A)}\ \frac{5}{4}\ km\qquad\textbf{(B)}\ \frac{35}{27}\ km\qquad\textbf{(C)}\ \frac{27}{20}\ km\qquad\textbf{(D)}\ \frac{7}{3}\ km\qquad\textbf{(E)}\ \frac{28}{9}\ km $

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely: Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: [b](1)[/b] Choose $k+1$ boxes; [b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. [b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step. [i]Proposed by Demetres Christofides, Cyprus[/i]

The currency exchange trades dinars (D), guilders (G), reals (R) and thalers (T). Exchange players have the right to make a purchase and sale transaction with each pair of currencies no more than once a day. The exchange rates are as follows: $D = 6G$,; $D=25R$, $D=120T$,$G = 4R$; $G=21T$, $R = 5T$. For example, the entry $D = 6G$ means that $1$ dinar can be bought for $6$ guilders (or $6$ guilders can be sold for $1$ dinar). In the morning the player had $32$ dinars. What is the maximum number that he can receive by evening a) in dinars? b) in thalers ?

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and $\vspace{0.1cm}$ $\hspace{1cm}ii) f(n)$ divides $n^3$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove that if $x, y$ and $n$ are positive integers such that $$x^{2024} + y^{2024} = 2^n,$$ then $x=y$.

Let $ABC$ be a triangle and let $M,N,P$ be points on the sides $BC$, $CA$ and $AB$ respectively such that \[ \frac{AP}{PB} = \frac{BM}{MC} = \frac{CN}{AN}. \] Prove that triangle if $MNP$ is equilateral then triangle $ABC$ is equilateral.

Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}$

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$ [asy] size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy] $\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

Let $n$ be a positive integer such that the sum of all its positive divisors (inclusive of $n$) equals to $2n + 1$. Prove that $n$ is an odd perfect square. related: https://artofproblemsolving.com/community/c6h515011 https://artofproblemsolving.com/community/c6h108341 (Putnam 1976) https://artofproblemsolving.com/community/c6h368488 https://artofproblemsolving.com/community/c6h445330 https://artofproblemsolving.com/community/c6h378928

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.