The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?

$\angle A,\angle B,\angle C$ are angles of a triangle such that $\sin^2A+\sin^2B=\sin^2C$, then $\angle C$ in degrees is equal to $\textbf{(A)}~30$ $\textbf{(B)}~90$ $\textbf{(C)}~45$ $\textbf{(D)}~\text{none of the above}$

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

[i]A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov[/i] A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark $2007$ points on this circle, after that Amayak removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick.

The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience. [list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$. [b](b)[/b] Determine all $n$ for which this is possible.[/list]

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.

In the land of Brobdingnag, a parking lot has $2020$ parking spaces in a row. $674$ cars, each two parking spaces wide, arrive at the parking lot one by one. Each car parks in a pair of consecutive vacant spaces, selected uniformly at random over all such pairs; for example, the first car can park in $2019$ ways, all with equal probability. If no pair of consecutive vacant spaces remain when a car arrives, it leaves disappointedly. What is the probability that all $674$ cars successfully park?

Initially, one cow is located on every negative integer on the number line. Each day, Farmer John chooses an integer $k$ where the interval $[k, k+5000]$ has cows. First, he moves each cow in $[k, k+5000]$ to another integer in the interval, so that no two cows move to the same integer. Then, he chooses a cow in the interval and removes it. Can Farmer John get a cow on $100,000,000$ after some time?

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ [i]Proposed by A. Smirnov[/i]

$\triangle ABC$ 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 $B$.