$\triangle ABC$ has side lengths $13$, $14$, and $15$. Let the feet of the altitudes from $A$, $B$, and $C$ be $D$, $E$, and $F$, respectively. The circumcircle of $\triangle DEF$ intersects $AD$, $BE$, and $CF$ at $I$, $J$, and $K$ respectively. What is the area of $\triangle IJK$?

Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$(Here $a_{0}=a_{n},a_{n+1}=a_{1}$). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$.

Let $a, b, c, x$ and $y$ be positive real numbers such that $ax + by \leq bx + cy \leq cx + ay$. Prove that $b \leq c$.

If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0); draw(A--E--B--F--C--D--A--F^^E--D--B); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, W); label("F", F, S); label("H", (.8,0.6)); label("I", (0.4,1.4)); [/asy] $ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{7}{15}\qquad\textbf{(D)}\ \frac{8}{15}\qquad\textbf{(E)}\ \frac{3}{5} $

When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: $\textbf{(A) }\tfrac12k(k-1)\qquad \textbf{(B) }\tfrac12k(k+1)\qquad \textbf{(C) }2k-1\qquad \textbf{(D) }2k\qquad \textbf{(E) }2k+1$

On a circle $\omega$ with center O and radius $r$ three different points $A, B$ and $C$ are chosen. Let $\omega_1$ and $\omega_2$ be the circles that pass through $A$ and are tangent to line $BC$ at points $B$ and $C$, respectively. (a) Show that the product of the areas of $\omega_1$ and $\omega_2$ is independent of the choice of the points $A, B$ and $C$. (b) Determine the minimum value that the sum of the areas of $\omega_1$ and $\omega_2$ can take and for what configurations of points $A, B$ and $C$ on $\omega$ this minimum value is reached.

Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear.

Let $AH$ be altitude in acute trinagle $ABC$, inscribed in circle $s$. Points $D$ and $E$ are chosen on segment $BH$. Points $X$ and $Y$ are chosen on rays $AD$ and $AE$, respectively, such that midpoints of segments $DX$ and $EY$ lies on $s$. Suppose that points $B$, $X$, $Y$ and $C$ are concyclic. Prove that $BD+BE=2CH$.

Do there exist distinct numbers $x,y,z$ from $[0,\dfrac{\pi}{2}]$, such that six number $\sin x$, $\sin y$,$\sin z$, $\cos x$, $\cos y$, $\cos z$ could be partitioned into 3 pairs with equal sums? [I]Proposed by A. Golovanov[/i]

What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?

There are several sets of three different numbers whose sum is $15$ which can be chosen from $\{ 1,2,3,4,5,6,7,8,9 \} $. How many of these sets contain a $5$? $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

A table $2 \times 2024$ is filled with positive integers. Specifically, the first row is filled with numbers from the set $\{1, \ldots, 2023\}$. It turned out that for any two columns the difference of numbers from the first row is divisible by the difference of numbers from the second row, while all numbers in the second row are pairwise different. Is it true for sure that the numbers in the first row are equal? Ivan Kukharchuk

Equilateral triangle $A_1A_2A_3$ has side length $15$ and circumcenter $M$. Let $N$ be a point such that $\angle A_3MN = 72^{\circ}$ and $MN = 7$. The circle with diameter $MN$ intersects lines $MA_1$, $MA_2$, and $MA_3$ again at $B_1$, $B_2$, and $B_3$, respectively. The value of $NB_1^2+NB_2^2+NB_3^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Tiebreaker #4[/i]

Given that \[A=\sum_{n=1}^\infty\frac{\sin(n)}{n},\] determine $\lfloor100A\rfloor$.

Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.

A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

Let $m$ be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements $1$, there exists a subset having the modulus of the sum of the elements in the subset greater than $m$. Prove that $$\frac14 \le m \le \frac12.$$ (Optional Task for 3p) Find a smaller value for the RHS.

There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme. [i]A. Magazinov[/i]

Two identical decks have 36 cards each. One deck is shuffled and put on top of the second. For each card of the top deck, we count the number of cards between it and the corresponding card of the bottom deck. What is the sum of these numbers? Sorry if this has been posted before but I would like to know if I solved it correctly. Thanks!

A positive integer $n$ is said to be $interesting$ if there exist some coprime positive integers $a$ and $b$ such that $n = a^2 - ab + b^2$. Show that if $n^2$ is $interesting$, then $n$ or $3n$ is $interesting$.

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$, $2$, $3$, $4$, such as $1234$, $1432$, or $3124$.

In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.

Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. [i]Ciprian Manolescu[/i]