The sides of a triangle are equal to $\sqrt2, \sqrt3, \sqrt4$ and its angles are $\alpha, \beta, \gamma$, respectively. Prove that the equation $x\sin \alpha + y\sin \beta + z\sin \gamma = 0$ has exactly one solution in integers $x, y, z$.

Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Given an $8 \times 8$ chess board. Each knight is allowed to move between two squares located at opposite vertices of $2 \times 3$ or $3 \times 2$ rectangles. There are four knights that move on the board, evenly start from the same cell $X$ and return to $X$ and then stop. Assume that every square on the chessboard has at least one of these four roosters moving through. Prove that there exists a square $Y$ that is different from $X$ such that it is moved over no less than twice by the same knight or by different knights.

Two players by turn paint the circles on the given picture each with his colour. At the end, the rest of the area of each of small triangles is painted by the colour of the majority of vertices of this triangle. The winner is one who gets larger area of his colour (the area of circles is taken into account). Does any of them have winning strategy? If yes, then who wins? \[ \begin{picture}(60,60) \put(5,3){\put(3,0){\line(6,0){8}} \put(17,0){\line(6,0){8}} \put(31,0){\line(6,0){8}} \put(45,0){\line(6,0){8}} \put(10,14){\line(6,0){8}} \put(24,14){\line(6,0){8}} \put(38,14){\line(6,0){8}} \put(17,28){\line(6,0){8}} \put(31,28){\line(6,0){8}} \put(24,42){\line(6,0){8}} \put(1,2){\line(1,2){5}} \put(15,2){\line(1,2){5}} \put(29,2){\line(1,2){5}} \put(43,2){\line(1,2){5}} \put(8,16){\line(1,2){5}} \put(22,16){\line(1,2){5}} \put(36,16){\line(1,2){5}} \put(15,30){\line(1,2){5}} \put(29,30){\line(1,2){5}} \put(22,44){\line(1,2){5}} \put(13,2){\line( \minus{} 1,2){5}} \put(27,2){\line( \minus{} 1,2){5}} \put(41,2){\line( \minus{} 1,2){5}} \put(55,2){\line( \minus{} 1,2){5}} \put(20,16){\line( \minus{} 1,2){5}} \put(34,16){\line( \minus{} 1,2){5}} \put(48,16){\line( \minus{} 1,2){5}} \put(27,30){\line( \minus{} 1,2){5}} \put(41,30){\line( \minus{} 1,2){5}} \put(34,44){\line( \minus{} 1,2){5}} \put(0,0){\circle{6}} \put(14,0){\circle{6}} \put(28,0){\circle{6}} \put(42,0){\circle{6}} \put(56,0){\circle{6}} \put(7,14){\circle{6}} \put(21,14){\circle{6}} \put(35,14){\circle{6}} \put(49,14){\circle{6}} \put(14,28){\circle{6}} \put(28,28){\circle{6}} \put(42,28){\circle{6}} \put(21,42){\circle{6}} \put(35,42){\circle{6}} \put(28,56){\circle{6}}} \end{picture}\]

Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$. [i](4 points)[/i]

Two players play the following game, using a round table $4$ feet in diameter, and a large pile of quarters. Each player can put in his turn one quarter on the table, but the one who cannot put a quarter (because there is no free space on the table) loses the game. Is there a winning strategy for the first or for the second player?

In a triangle $ABC$, the circle through $C$ touching $AB$ at $A$ and the circle through $B$ touching $AC$ at $A$ have different radii and meet again at $D$. Let $E$ be the point on the ray $AB$ such that $AB = BE$. The circle through $A$, $D$, $E$ intersect the ray $CA$ again at $F$ . Prove that $AF = AC$.

The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.

How many three-digit positive integers have digits which sum to a multiple of $10$?

Find all the positive integers $n$ such that: $(1)$ $n$ has at least $4$ positive divisors. $(2)$ if all positive divisors of $n$ are $d_1,d_2,\cdots ,d_k,$ then $d_2-d_1,d_3-d_2,\cdots ,d_k-d_{k-1}$ form a geometric sequence.

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

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

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

p1. $A$ is a set of numbers. The set $A$ is closed to subtraction, meaning that the result of subtracting two numbers in $A$ will be returns a number in $A$ as well. If it is known that two members of $A$ are $4$ and $9$, show that: a. $0\in A$ b. $13 \in A$ c. $74 \in A$ d. Next, list all the members of the set $A$ . p2. $(2, 0, 4, 1)$ is one of the solutions/answers of $x_1+x_2+x_3+x_4=7$. If all solutions belong on the set of not negative integers , specify as many possible solutions/answers from $x_1+x_2+x_3+x_4=7$ p3. Adi is an employee at a textile company on duty save data. One time Adi was asked by the company leadership to prepare data on production increases over five periods. After searched by Adi only found four data on the increase, namely $4\%$, $9\%$, $7\%$, and $5\%$. One more data, namely the $5$th data, was not found. Investigate increase of 5th data production, if Adi only remembers that the arithmetic mean and median of the five data are the same. p4. Find all pairs of integers $(x,y)$ that satisfy the system of the following equations: $$\left\{\begin{array}{l} x(y+1)=y^2-1 \\ y(x+1)=x^2-1 \end{array} \right. $$ p5. Given the following image. $ABCD$ is square, and $E$ is any point outside the square $ABCD$. Investigate whether the relationship $AE^2 + CE^2 = BE^2 +DE^2$ holds in the picture below. [img]https://cdn.artofproblemsolving.com/attachments/2/5/a339b0e4df8407f97a4df9d7e1aa47283553c1.png[/img]

Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$. a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent. b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.

Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersections of some line $e$ with the sides $AB, CD$ and $AD, BC$, respectively (or their extensions). Given the points $M, N, P, Q$ and the length $p$ of side $AB$, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$

We construct similar isosceles triangles on the sides of the triangle $ ABC $: triangle $ APB $ outside the triangle $ ABC $ ($ AP = PB $), triangle $ CQA $ outside the triangle $ ABC $ ($ CQ = QA $), triangle $ CRB $ inside the triangle $ ABC $ ($ CR = RB $). Prove that $ APRQ $ is a parallelogram or that the points $ A, P, R, Q $ lie on a straight line.

The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$? [asy] size((100)); draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(9,9)--(0,3)--cycle); dot((3,0)); dot((0,3)); dot((9,9)); dot((0,0)); dot((9,0)); dot((0,9)); label("$A$", (0,9), NW); label("$B$", (9,9), NE); label("$C$", (9,0), SE); label("$D$", (3,0), S); label("$E$", (0,0), SW); label("$F$", (0,3), W); [/asy] $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{5}{18}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{7}{20} $

Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.

Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least $100$ml of milk, and drink half of it. (2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle. Whoever cannot do any operations loses the game. Who has a winning strategy? [i] Proposed by Chu-Lan Kao and usjl[/i]

For $a, b, c>0,$ and $k\geq1,$ prove that \[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\] Author: MIHALY BENCZE