Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$. [i]Gheorghe Szolosy[/i]

At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.

Right $\vartriangle ABC$ has side lengths $6, 8$, and $10$. Find the positive integer $n$ such that the area of the region inside the circumcircle but outside the incircle of $\vartriangle ABC$ is $n\pi$. [img]https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png[/img]

Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$. If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions: $$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$ for all real numbers $ x,y,t $ with $ t\in [0,1] . $ Prove that: [b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $ [b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds. $$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

Find all solution $(p,r)$ of the "Pythagorean-Euler Theorem" $$p^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^p$$Where $p$ is a prime and $r$ is a positive integer. [i]Proposed by Li4 and Untro368[/i]

let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $

The sequence $a_1, a_2,...$ is defined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.

Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality: \[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]

Let $C$ be one of the two points of intersection of circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$, respectively. The line $O_1O_2$ intersects the circles at points $A$ and $B$ as shown in the figure. Let $K$ be the second point of intersection of line $AC$ with circle $\omega_2$, $L$ be the second point of intersection of line $BC$ with circle $\omega_1$. Lines $AL$ and $BK$ intersect at point $D$. Prove that $AD=BD$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/6/4/2cdccb43743fcfcb155e846a0e05ec79ba90e4.png[/img]

The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression $$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $ a_1,a_2,...,a_{2003}\geq 0$, such that $ a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2$ and $ a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1$. Determine the minimum and maximum value of $ a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2$.

Find the number of permutations of the letters $AAAABBBCC$ where no letter is next to another letter of the same type. For example, count $ABCABCABA$ and $ABABCABCA$ but not $ABCCBABAA$.

Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1=AC\cap FB$, $B_1=BD\cap AC$, $C_1=CE\cap BD$, $D_1=DF\cap CE$, $E_1=EA\cap DF$, and $F_1=FB\cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parrallel. (Here, we use $l_1\cap l_2$ to denote the intersection point of lines $l_1$ and $l_2$)

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

Find the distance $\overline{CF}$ in the diagram below where $ABDE$ is a square and angles and lengths are as given: [asy] markscalefactor=0.15; size(8cm); pair A = (0,0); pair B = (17,0); pair E = (0,17); pair D = (17,17); pair F = (-120/17,225/17); pair C = (17+120/17, 64/17); draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle); label("$A$", A, S); label("$B$", B, S); label("$C$", C, dir(0)); label("$D$", D, N); label("$E$", E, N); label("$F$", F, W); label("$8$", (F+E)/2, NW); label("$15$", (F+A)/2, SW); label("$8$", (C+B)/2, SE); label("$15$", (D+C)/2, NE); draw(rightanglemark(E,F,A)); draw(rightanglemark(D,C,B)); [/asy] The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?

Suppose that a bounded subset $ S$ of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of $ S$ can be covered by a finite number of rectifiable arcs. [i]L. Geher[/i]

Find all angles a for which the set of numbers $\sin a$, $\sin 2a$, $\sin 3a$ coincides with the set $cos a$, $cos 2a$, $cos 3a$.

A teacher writes the positive integers from $1$ to $12$ on a blackboard. Every minute, they choose a number $k$ uniformly at random from the written numbers, subtract $k$ from each number $n \geq k$ on the blackboard (without touching the numbers $n<k$), and erase every $0$ on the board. Estimate the expected number of minutes that pass before the board is empty. An estimate of $E$ earns $2^{1-0.5|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.2[/i]