Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

We say that a set $S$ of $3$ unit squares is \textit{commutable} if $S = \{s_1,s_2,s_3\}$ for some $s_1,s_2,s_3$ where $s_2$ shares a side with each of $s_1,s_3$. How many ways are there to partition a $3\times 3$ grid of unit squares into $3$ pairwise disjoint commutable sets? [i]Proposed by Srinivasan Sathiamurthy[/i]

There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]

Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that \[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\] Prove that the points $E,F,G,H,K,L$ all lie on a sphere.

In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively. Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ . [i]Proposed by Hojoo Lee, Korea[/i]

A subset $S$ of $\{1, 2, 3, \ldots, 2025\}$ is called [i]balanced[/i] if for all elements $a$ and $b$ both in $S,$ there exists an element $c$ in $S$ such that $2025$ divides $a+b-2c.$ Compute the number of [i]nonempty[/i] balanced sets.

$6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} = $ $ \textbf{(A)}\ 6^{6}\qquad\textbf{(B)}\ 6^{7}\qquad\textbf{(C)}\ 36^{6}\qquad\textbf{(D)}\ 6^{36}\qquad\textbf{(E)}\ 36^{36} $

Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer. a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice. b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?

Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.

In the convex hexagon $ABCDEF, AB, BC$ and $CD$ are respectively parallel to $DE, EF$ and $FA$. If $AB = DE$, prove that $BC = EF$ and $CD = FA$.

Let $O$ denote the origin and let $\gamma$ be the circle with center $(1,0)$ and radius $1$ in the Cartesian system of coordinates. Let $\lambda$ be a real number from the interval $(0,2)$, and let the line $x=\lambda$ intersect the circle $\gamma$ at points $P$ and $Q$. The lines $OP$ and $OQ$ intersect the line $x=2-\lambda$ at the points $P'$ and $Q'$, respectively. Let $\mathcal G$ denote the locus of such points $P'$ and $Q'$ as $\lambda$ varies over the interval $(0,2)$. Prove that there exist points $R$ and $S$ different from the origin in the plane such that for every $A\in \mathcal G$ there exists a point $A'$ on line $OA$ satisfying \[ A'R^2=(A'S-OS)^2=A'A\cdot A'O.\] [i]Proposed by: Áron Bán-Szabó, Budapest[/i]

The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$ at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point $C'$. Points $A'$ and $B'$ are defined similarly. Prove that $A', B'$ and $C'$ lie on one line perpendicular to the line $MH$.

Several napkins of equal size and of shape of a unit disc were placed on a table (with overlappings). Is it always possible to hammer several point-sized nails so that all the napkins will be thus attached to the table with the same number of nails? (The nails cannot be hammered into the borders of the discs). Vladimir Dolnikov, Pavel Kozhevnikov

There are $12$ points that are vertices of a regular polygon with $12$ sides. Rafael must draw segments that have their two ends at two of the points drawn. He is allowed to have each point be an endpoint of more than one segment and for the segments to intersect, but he is prohibited from drawing three segments that are the three sides of a triangle in which each vertex is one of the $12$ starting points. Find the maximum number of segments Rafael can draw and justify why he cannot draw a greater number of segments.

Eric plans to compete in a triathlon. He can average $ 2$ miles per hour in the $ \tfrac{1}{4}$-mile swim and $ 6$ miles per hour in the $ 3$-mile run. His goal is to finish the triathlon in $ 2$ hours. To accomplish his goal what must his average speed, in miles per hour, be for the $ 15$-mile bicycle ride? $ \textbf{(A)}\ \frac{120}{11} \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ \frac{56}{5} \qquad \textbf{(D)}\ \frac{45}{4} \qquad \textbf{(E)}\ 12$

$ADC$ and $ABC$ are triangles such that $AD=DC$ and $AC=AB$. If $\angle CAB=20^{\circ}$ and $\angle ADC =100^{\circ}$, without using Trigonometry, prove that $AB=BC+CD$.

Prove that for every integer $k \ge 2$ there are $k$ different natural numbers $n_1$, $n_2$, $...$ , $n_k$ such that: $$\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}=\frac{3}{17}$$

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true: \[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\] [i]Maxim Didin[/i]

[b]R[/b]thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?

Given positive integers $m$ and $n \ge m$, determine the largest number of dominoes ($1\times2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2n$ columns consisting of cells ($1 \times 1$ squares) so that: (i) each domino covers exactly two adjacent cells of the board; (ii) no two dominoes overlap; (iii) no two form a $2 \times 2$ square; and (iv) the bottom row of the board is completely covered by $n$ dominoes.

Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form equilateral triangles by drawing an edge between every pair of neighboring squares. If this figure is continued symmetrically off to infinity, what is the ratio between the number of triangles and the number of squares?

Let $ABC$ be a triangle, and let $D$, $E$, and $F$ be the midpoints of sides $BC$, $CA$, and $AB$, respectively. Let the angle bisectors of $\angle FDE$ and $\angle FBD$ meet at $P$. Given that $\angle BAC = 37^o$ and $\angle CBA = 85^o$ determine the degree measure of $\angle BPD$.

Let $ABC$ be a right-angled triangle with $\angle{B}=90^{\circ}$. Let $BD$ is the altitude from $B$ on $AC$. Let $P,Q$ and $I $be the incenters of triangles $ABD,CBD$ and $ABC$ respectively.Show that circumcenter of triangle $PIQ$ lie on the hypotenuse $AC$.