Do there exist infinitely many triples of positive integers $(a, b, c)$ such that $a$, $b$, $c$ are pairwise coprime, and $a! + b! + c!$ is divisible by $a^2 + b^2 + c^2$? [i]Proposed by Anzo Teh Zhao Yang[/i]

Mr. Squash bought a large parking lot in Utah, which has an area of $600$ square meters. A car needs $6$ square meters of parking space while a bus needs $30$ square meters of parking space. Mr. Squash charges $\$2.50$ per car and $\$7.50$ per bus, but Mr. Squash can only handle at most $60$ vehicles at a time. Find the ordered pair $(a,b)$ where $a$ is the number of cars and $b$ is the number of buses that maximizes the amount of money Mr. Squash makes. [i]Proposed by Nathan Cho[/i]

Let $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$, and $(x_5, y_5)$ be the vertices of a regular pentagon centered at $(0, 0)$. Compute the product of all positive integers k such that the equality $x_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k$ must hold for all possible choices of the pentagon.

For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm. (translated by Miklós Maróti)

Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.

Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$

Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. [i](Proposed by Bogdan Rublov)[/i]

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$

Points $A_{1}, B_{1}, C_{1}$ are given inside an equilateral triangle $ABC$ such that $\widehat{B_{1}AB}= \widehat{A1BA}= 15^{0}, \widehat{C_{1}BC}= \widehat{B_{1}CB}= 20^{0}, \widehat{A_{1}CA}= \widehat{C_{1}AC}= 25^{0}$. Find the angles of triangle $A_{1}B_{1}C_{1}$.

How many ways are there to place $31$ knights in the cells of an $8 \times 8$ unit grid so that no two attack one another? (A knight attacks another knight if the distance between the centers of their cells is exactly $\sqrt5$.)

The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? $\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13} $

Fine Hall has a broken elevator. Every second, it goes up a floor, goes down a floor, or stays still. You enter the elevator on the lowest floor, and after $8$ seconds, you are again on the lowest floor. If every possible such path is equally likely to occur, the probability you experience no stops is $\tfrac{a}{b},$ where $a,b$ are relatively prime positive integers. Find $a + b.$

Let be a natural number $ n. $ Calculate $$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} . $$ Here, $ \# $ means cardinal.

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[f(x(x + f(y))) = (x + y)f(x),\] for all $x, y \in\mathbb{R}$.

If $z\in\mathbb{C}$ is a solution of the equation $$x^n+a_1x^{n-1}+a_2x^{n-2}+\ldots+a_n=0$$ with real coefficients $0<a_n\leq a_{n-1}\leq\ldots\leq a_1<1$, show that $|z|<1$.

Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$

In a summer camp that is going to last $n$ weeks, you want to divide the time into $3$ periods so that each period starts on a Monday and ends on a Sunday. The first period will be dedicated to artistic work, the second will be for sports and in the third there will be a technological workshop. During each term, a Monday will be chosen for an expert on the topic of the term to give a talk. Let $C(n)$ be the number of ways in which the activity calendar can be made. (For example, if $n=10$ one way the calendar could be done is by putting the first four weeks for art and the artist talk on the first Monday; the next $5$ weeks could be for sports, with the athlete visit on the fourth Monday of that period; the remaining week would be for the technology workshop and the talk would be on Monday of that week.) Calculate $C(8)$.

Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$. [16 points out of 100 for the 6 problems]

Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$. Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.

Let $a, b, c$ be positive real numbers with $abc = 1$. Find all possible values ​​of the expression $$\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}$$ can take.

Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

Margie bought $3$ apples at a cost of $50$ cents each. She paid with a $5$-dollar bill. How much change did Margie receive? $ \textbf{(A)}\$1.50\qquad\textbf{(B)}\$2.00\qquad\textbf{(C)}\$2.50\qquad\textbf{(D)}\$3.00\qquad\textbf{(E)}\$3.50 $