Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.

Let $ABC$ be a non-right-angled triangle and let $H$ be its orthocenter. Let $M_1,M_2,M_3$ be the midpoints of the sides $BC$, $CA$, $AB$ respectively. Let $A_1$, $B_1$, $C_1$ be the symmetrical points of $H$ with respect to $M_1$, $M_2$ and $M_3$ respectively, and let $A_2$, $B_2$, $C_2$ be the orthocenters of the triangles $BA_1C$, $CB_1A$ and $AC_1B$ respectively. Prove that: a) triangles $ABC$ and $A_2B_2C_2$ have the same centroid; b) the centroids of the triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ form a triangle similar with $ABC$.

Determine all triples $(a,b,c)$ of real numbers such that $$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$

Given is a positive integer $n$. Let $A$ be the set of points $x \in (0;1)$ such that $|x-\frac{p} {q}|>\frac{1}{n^3}$ for each rational fraction $\frac{p} {q}$ with denominator $q \leq n^2$. Prove that $A$ is a union of intervals with total length not exceeding $\frac{100}{n}$. Proposed by Fedor Petrov

Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

In a triangle $ABC$, the nine-point circle $(N)$ is tangent to the incircle $(I)$ and three excircles $(I_a), (I_b), (I_c)$ at the Feuerbach points $F, F_a, F_b, F_c$. Tangents of $(N)$ at $F, F_a, F_b, F_c$ bound a quadrangle $PQRS$. Show that the Euler line of $ABC$ is a Newton line of $PQRS$. Luis González

Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of $ 4$ to $ 3$. His wife got twice as much as the son. If the cook received a bequest of $ \$500$, then the entire estate was: $ \textbf{(A)}\ \$3500 \qquad\textbf{(B)}\ \$5500 \qquad\textbf{(C)}\ \$6500 \qquad\textbf{(D)}\ \$7000 \qquad\textbf{(E)}\ \$7500$

Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.

Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ [i]rich[/i] if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$. For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$, every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?

Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.

Describe the region $R$ consisting of the points $(a,b)$ of the cartesian plane for which both (possibly complex) roots of the polynomial $z^2+az+b$ have absolute value smaller than $1$.

On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?

Prove that there are infinitely many positive integers $n$ such that $$3^{(n-2)^{n-1}-1} -1\vdots 17n^2$$ (I. Bliznets)

In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9. While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonder which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)

There is a chess tournament with $2n$ players ($n > 1$). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most $n^2$ matches. Conversely, show that if there are at most $n^2$ matches, then it is possible to arrange them so that we cannot find three players who all play each other.

Given are $2021$ prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with $6$ and $12$. Prove that there are at least two equal numbers.

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Show that the triangles $ABH,BCH$ and $CAH$ have the same perimeter if and only if the triangle $ABC$ is equilateral.

For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?

Prove that among any $10$ entries of the table $$0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, ... \,\,\,\, 9$$ $$9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, 2 \,\,\,\, ... \,\,\,\, 8$$ $$8 \,\,\,\, 9 \,\,\,\, 0 \,\,\,\, 1 \,\,\,\, ... \,\,\,\, 7$$ $$1 \,\,\,\, 2 \,\,\,\, 3 \,\,\,\, 4 \,\,\,\, ... \,\,\,\, 0$$ standing in different rows and different columns, at least two are equal. (A Savin)

How many ways are there to make change for $55$ cents using any number of pennies nickles, dimes, and quarters? $\text{(A) }42\qquad\text{(B) }49\qquad\text{(C) }55\qquad\text{(D) }60\qquad\text{(E) }78$

Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of a triangle $ABC.$ If the point $E$ on the ray $BA$ and the point $F$ on the ray $CA$ satisfy the condition \[BE=CF=\frac{AB+BC+CA}{2}\] show that the lines $EF$ and $DI$ are perpendicular.

Bisectors of $\angle BAC$, $\angle CAD$ in a rectangle $ABCD$ , intersect the sides $BC$, $CD$ at points $M$ and $N$ resp. Prove that $\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1$