Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation $ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$ has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$

Find all real solutions of the system $\begin{cases} x^3 + y^3 = 1 \\ x^4 + y^4 = 1 \end{cases}$

Let $ \mathcal{H}$ be a family of finite subsets of an infinite set $ X$ such that every finite subset of $ X$ can be represented as the union of two disjoint sets from $ \mathcal{H}$. Prove that for every positive integer $ k$ there is a subset of $ X$ that can be represented in at least $ k$ different ways as the union of two disjoint sets from $ \mathcal{H}$. [i]P. Erdos[/i]

The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$. (A Zaslavsky)

Let there be an integer $n\geq2$. In a chess tournament $n$ players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$ such that for every $i (1\leq i\leq n-1)$ the player $P_i$ won against player $P_{i+1}$.

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$

Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n - 24)!$ such that no two distinct divisors $s, t$ of the same color satisfy $s | t$.

Find the smallest positive integer $n$ such that the equation $\sqrt3 z^{n+1} - z^n - 1 = 0$ has a root on the unit circle.

Prove that if the product of positive numbers $a,b$ and $c$ equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$

(a) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red.

Find all positive integers $n$ such that $4^n + 4n + 1$ is a perfect square.

$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $

Show that it is possible to circumscribe a circle of radius $R$ about, and inscribe a circle of radius $r$ in some triangle with one angle equal to $a$, if and only if $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$

On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$, when the hour hand and the minute hannd are at right angles. The difference [i]in minutes[/i] between these two instants is written as $a + \dfrac{b}{c}$, where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a+b+c$?

The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

[u]Round 5[/u] [b]p13.[/b] Simplify $\frac11+\frac13+\frac16+\frac{1}{10}+\frac{1}{15}+\frac{1}{21}$. [b]p14.[/b] Given that $x + y = 7$ and $x^2 + y^2 = 29$, what is the sum of the reciprocals of $x$ and $y$? [b]p15.[/b] Consider a rectangle $ABCD$ with side lengths $AB = 3$ and $BC = 4$. If circles are inscribeδ in triangles $ABC$ and $BCD$, how far are the centers of the circles from each other? [u]Round 6[/u] [b]p16.[/b] Evaluate $\frac{2!}{1!} +\frac{3!}{2!} +\frac{4!}{3!} + ... +\frac{99!}{98!}+\frac{100!}{99!}$ . [b]p17.[/b] Let $ABCD$ be a square of side length $2$. A semicircle is drawn with diameter $\overline{AC}$ that passes through point $B$. Find the area of the region inside the semicircle but outside the square. [b]p18.[/b] For how many positive integer values of $k$ is $\frac{37k - 30}{k}$ a positive integer? [u]Round 7[/u] [b]p19.[/b] Two parallel planar slices across a sphere of radius $25$ create cross sections of area $576\pi$ and $225\pi$. What is the maximum possible distance between the two slices? [b]p20.[/b] How many positive integers cannot be expressed in the form $3\ell + 4m + 5t$, where $\ell$, $m$, and $t$ are nonnegative integers? [b]p21.[/b] In April, a fool is someone who is fooled by a classmate. In a class of $30$ students, $14$ people were fooled by someone else and $29$ people fooled someone else. What is the largest positive integer $n$ for which we can guarantee that at least one person was fooled by at least $n$ other people? [u]Round 8[/u] [b]p22.[/b] Let $$S = 4 + \dfrac{12}{4 +\dfrac{ 12}{4 +\dfrac{ 12}{4+ ...}}}.$$ Evaluate $4 +\frac{ 12}{S}.$ [b]p23.[/b] Jonathan is buying bananagram sets for $\$11$ each and flip-flops for $\$17$ each. If he spends $\$227$ on purchases for bananagram sets and flip-flops, what is the total number of bananagram sets and flip-flops he bought? [b]p24.[/b] Alan has a $3 \times 3$ array of squares. He starts removing the squares one at a time such that each time he removes one square, all remaining squares share a side with at least two other remaining squares. What is the maximum number of squares Alan can remove? PS. You should use hide for answers. Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$. [i]Proposed by Luke Robitaille[/i]

Let $f(x)=cx(x-1)$, where $c$ is a positive real number. We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^n(x)$ are real. What is the smallest possible value of $c$?

A sequence of real numbers $\{x_n\}$ satisfies the recursion $x_{n+1} = 4x_n - 4x^2_n$, where $n \ge 1$. If $x_{2023} = 0$, compute the number of distinct possible values for $x_1$.

Let $ z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) $. The sum $$ \sum_{k = 0}^{13} \dfrac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} $$ can be written in the form $ a - bi $. Find $ a + b $.

On an island live $n$ ($n \ge 2$) $xyz$s. Any two $xyz$s are either friends or enemies. Every $xyz$ wears a necklace made of colored beads such that any two $xyz$s that are befriended have at least one bead of the same color and any two $xyz$s that are enemies do not have any common colors in their necklaces. It is also possible for some necklaces not to have any beads. What is the minimum number of colors of beads that is sufficient to manufacture such necklaces regardless on the relationship between the $xyz$s?

Let $S =\{1, 2,..., 6\}$. How many functions $f : S \to S$ are there such that for all $s \in S$, $$f^5(s) = f(f(f(f(f(s))))) = 1?$$

As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded? [asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)); draw((2.45,.95)--(2.05,.55)); draw((0,2)--(4,2)--(4,3)--(0,3)--cycle); draw((2,2)--(2,3),dashed); draw((1.3,2.1)..(2,2.3)..(2.7,2.1),EndArrow); draw((1.3,3.1)..(2,3.3)..(2.7,3.1),EndArrow); draw((0,4)--(4,4)--(4,6)--(0,6)--cycle); draw((0,5)--(4,5),dashed); draw((-.1,4.3)..(-.3,5)..(-.1,5.7),EndArrow); draw((3.9,4.3)..(3.7,5)..(3.9,5.7),EndArrow);[/asy] [asy] unitsize(5); draw((0,0)--(16,0)--(16,8)--(0,8)--cycle); draw((0,4)--(16,4),dashed); draw((8,0)--(8,8),dashed); draw(circle((1,3),.9)); draw(circle((7,7),.9)); draw(circle((15,5),.9)); draw(circle((9,1),.9)); draw((24,0)--(40,0)--(40,8)--(24,8)--cycle); draw((24,4)--(40,4),dashed); draw((32,0)--(32,8),dashed); draw(circle((31,1),.9)); draw(circle((33,1),.9)); draw(circle((31,7),.9)); draw(circle((33,7),.9)); draw((48,0)--(64,0)--(64,8)--(48,8)--cycle); draw((48,4)--(64,4),dashed); draw((56,0)--(56,8),dashed); draw(circle((49,1),.9)); draw(circle((49,7),.9)); draw(circle((63,1),.9)); draw(circle((63,7),.9)); draw((72,0)--(88,0)--(88,8)--(72,8)--cycle); draw((72,4)--(88,4),dashed); draw((80,0)--(80,8),dashed); draw(circle((79,3),.9)); draw(circle((79,5),.9)); draw(circle((81,3),.9)); draw(circle((81,5),.9)); draw((96,0)--(112,0)--(112,8)--(96,8)--cycle); draw((96,4)--(112,4),dashed); draw((104,0)--(104,8),dashed); draw(circle((97,3),.9)); draw(circle((97,5),.9)); draw(circle((111,3),.9)); draw(circle((111,5),.9)); label("(A)",(8,10),N); label("(B)",(32,10),N); label("(C)",(56,10),N); label("(D)",(80,10),N); label("(E)",(104,10),N);[/asy]