Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

Let be three real numbers $ a,b,c\in [0,1] $ satisfying the condition $ ab+bc+ca=1. $ Prove that $$ a^2+b^2+c^2\le 2, $$ and determine the cases in which equality is attained.

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.

Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.

Solve in natural numbers: $$x^3-y^3=xy+61$$

Given a quadrilateral $ABCD$ with $\angle ABC = \angle ADC$. Let $BM$ be the altitude of the triangle $ABC$, and $M$ belongs to $AC$. Point $M'$ is marked on the diagonal $AC$ so that $$\frac{AM \cdot CM'}{ AM' \cdot CM}= \frac{AB \cdot CD }{ BC \cdot AD}$$ Prove that the intersection point of $DM'$ and $BM$ coincides with the orthocenter of the triangle $ABC$. (M. Zhikhovich)

\[\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

Let $\vartriangle ABC$ be a triangle, and let $C_0, B_0$ be the feet of perpendiculars from $C$ and $B$ onto $AB$ and $AC$ respectively. Let $\Gamma$ be the circumcircle of $\vartriangle ABC$. Let E be a point on the $\Gamma$ such that $AE \perp BC$. Let $M$ be the midpoint of $BC$ and let $G$ be the second intersection of EM and $\Gamma$. Let $T$ be a point on $\Gamma$ such that $T G$ is parallel to $BC$. Prove that $T, A, B_0, C_0$ are concyclic.

A triangle \( ABC \) is given with centers \( O \) and \( I \) of the circumscribed and inscribed circles, respectively. Point \( A_1 \) is the reflection of \( A \) with respect to \( I \). Point \( A_2 \) is such that lines \( BA_1 \) and \( BA_2 \) are symmetric with respect to \( BI \), and lines \( CA_1 \) and \( CA_2 \) are symmetric with respect to \( CI \). Prove that \( AO^2 = |A_2O^2 - A_2I^2| \).

Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$.

Trapezoid $ABCD,$ with $AB \parallel CD,$ has side lengths $AB=11, BC=8, CD=19,$ and $DA=4.$ Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle{ABC}, \triangle{BCD}, \triangle{CDA},$ and $\triangle{DAB}.$

Evaluate $1 + 2 + 4 + 7$ $\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

A can contains a mixture of two liquids A and B in the ratio $7:5$. When $9$ litres of the mixture are drawn and replaced by the same amount of liquid $B$, the ratio of $A$ and $B$ becomes $7:9$. How many litres of liquid A was contained in the can initially? $\textbf{(A)}~18$ $\textbf{(B)}~19$ $\textbf{(C)}~20$ $\textbf{(D)}~\text{None of the above}$

Let $a,b,c$ be real numbers for which $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximal value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$.

$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.

Let $n>1$ be a given integer. In a deck of cards the cards are of $n$ different suites and $n$ different values, and for each pair of a suite and a value there is exactly one such card. We shuffle the deck and distribute the cards among $n$ players giving each player $n$ cards. The players' goal is to choose a way to sit down around a round table so that they will be able to do the following: the first player puts down an arbitrary card, and then each consecutive player puts down a card that has a different suite and different value compared to the previous card that was put down on the table. For which $n$ is it possible that the cards were distributed in such a way that the players cannot achieve their goal? (The players work together, and they can see each other's cards.) Proposed by [i]Anett Kocsis[/i], Budapest

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.

Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$ [i]Proposed by Adam Bertelli[/i]

What is the difference between the median and the mean of the following data set: $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$?

Find all real $x,y$ such that \[x^2 + 2y^2 + \frac{1}{2} \le x(2y+1) \]