$\boxed{A1}$ Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$

The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

Define $f(n)$ to be the sum of positive integers $k$ less than or equal to $n$ such that $\gcd(n, k)$ is prime. Find $f(2024)$.

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.

Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be $ \text{(A)}\ 10\qquad\text{(B)}\ 11\qquad\text{(C)}\ 19\qquad\text{(D)}\ 20\qquad\text{(E)}\ 25 $

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$

The circumcentre $O$ of a given cyclic quadrilateral $ABCD$ lies inside the quadrilateral but not on the diagonal $AC$. The diagonals of the quadrilateral intersect at $I$. The circumcircle of the triangle $AOI$ meets the sides $AD$ and $AB$ at points $P$ and $Q$, respectively; the circumcircle of the triangle $COI$ meets the sides $CB$ and $CD$ at points $R$ and $S$, respectively. Prove that $PQRS$ is a parallelogram.

If $r$ is a rational number, let $f(r) = \left( \frac{1-r^2}{1+r^2} , \frac{2r}{1+r^2} \right)$. Then the images of $f$ forms a curve in the $xy$ plane. If $f(1/3) = p_1$ and $f(2) = p_2$, what is the distance along the curve between $p_1$ and $p_2$?

Let real angles $\theta_1, \theta_2, \theta_3, \theta_4$ satisfy \begin{align*} \sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4 &= 0, \\ \cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4 &= 0. \end{align*} If the maximum possible value of the sum \[\sum_{i<j}\sqrt{1-\sin\theta_i\sin\theta_j-\cos\theta_i\cos\theta_j}\] for $i, j \in \{1, 2, 3, 4\}$ can be expressed as $a+b\sqrt{c}$, where $c$ is square-free and $a,b,c$ are positive integers, find $a+b+c$ [i]Proposed by Alex Li[/i]

Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal. [i](Karl Czakler)[/i]

Let $ ABCD$ be a convex quadrtilateral such that $ AB$ is not parallel with $ CD$. Let $ \Gamma_1$ be a circle that passes through $ A$ and $ B$ and is tangent to $ CD$ at $ P$. Also, let $ \Gamma_2$ be a circle that passes through $ C$ and $ D$ and is tangent to $ AB$ at $ Q$. Let the circles $ \Gamma_1$ and $ \Gamma_2$ intersect at $ E$ and $ F$. Prove that $ EF$ passes through the midpoint of $ PQ$ iff $ BC \parallel AD$.

[b]8.1 / 7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ? [b]8.2 / 7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]8.3 / 7.6[/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. [b]8.4[/b] Prove that the sum of all divisors of the number $n^2$ is odd. [b]8.5[/b] A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle. [b]8.6[/b] Numbers $x_1, x_2, . . . $ are constructed according to the following rule: $$x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...$$ Prove that no matter how much we continued this construction, all the resulting numbers will be no less $1/5$ and no more than $2$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.

Given positive integer $n \geq 2$. Now Cindy fills each cell of the $n*n$ grid with a positive integer not greater than $n$ such that the numbers in each row are in a non-decreasing order (from left to right) and numbers in each column is also in a non-decreasing order (from top to bottom). We call two adjacant cells form a $domino$ , if they are filled with the same number. Now Cindy wants the number of $domino$s as small as possible. Find the smallest number of $dominos$ Cindy can reach. (Here, two cells are called adjacant if they share one common side)

In an acute triangle $ABC$ the heights $AD, BE$ and $CF$ intersecting at $H{}$. Let $O{}$ be the circumcenter of the triangle $ABC$. The tangents to the circle $(ABC)$ drawn at $B{}$ and $C{}$ intersect at $T{}$. Let $K{}$ and $L{}$ be symmetric to $O{}$ with respect to $AB$ and $AC$ respectively. The circles $(DFK)$ and $(DEL)$ intersect at a point $P{}$ different from $D{}$. Prove that $P, D$ and $T{}$ lie on the same line. [i]Proposed by Don Luu (Vietnam)[/i]

A six place number is formed by repeating a three place number; for example, $ 256256$ or $ 678678$, etc. Any number of this form is always exactly divisible by: $ \textbf{(A)}\ 7 \text{ only} \qquad\textbf{(B)}\ 11 \text{ only} \qquad\textbf{(C)}\ 13 \text{ only} \qquad\textbf{(D)}\ 101 \qquad\textbf{(E)}\ 1001$

What is the area of the shaded pinwheel shown in the $5\times 5$ grid? [asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); }[/asy] $\textbf{(A)}\: 4\qquad \textbf{(B)}\: 6\qquad \textbf{(C)}\: 8\qquad \textbf{(D)}\: 10\qquad \textbf{(E)}\: 12$

Consider the positive reals $ x$, $ y$ and $ z$. Prove that: a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$. b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.

Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for a) $n=6$ b) $n=1986$ ?

Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by \begin{align*} f(0) &= 1\\ f(2x) &= \lfloor\phi f(x)\rfloor\\ f(2x+1) &= f(2x) + f(x). \end{align*} Find the remainder when $f(2007)$ is divided by $2008$.

How many ways are there to fill a $3\times3\times6$ rectangular prism with $1\times1\times2$ blocks? Rotations are not distinct. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\max\left(2\left(1-\left|\frac{C-A}{C}\right|\right),0\right)$. [i]2017 CCA Math Bonanza Lightning Round #5.3[/i]