Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

The quadrilateral ABCD is inscribed in a circle which has diameter BD. Points A’ and B’ are symmetric to A and B with respect to the line BD and AC respectively. If the lines A’C, BD intersect at P and AC, B’D intersect at Q, prove that PQ is perpendicular to AC.

Find all possible values of $n$ for which a rectangular board $9\times n$ can be partitioned into tiles of the shape: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8wLzdjM2Y4ZmE0Zjg1YWZlZGEzNTQ1MmEyNTc3ZjJkNzBlMjExYmY1LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yMiBhdCA1LjEzLjU3IEFNLnBuZw==[/img]

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are positive integers. What is $a+b$? [i]2021 CCA Math Bonanza Individual Round #12[/i]

Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.

Let \[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\] Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$. For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times. Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.

The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors. i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one). ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times. iii) Prove that if however $M=2012$ then $n \leq 1007$. ([i]Dan Schwarz[/i])

On an infinite sheet of tiles, an infinite number of $1 \times 2$ tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with $1 \times 2$ checkered rectangles? [hide=original wording]Uz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem? [/hide]

At a party with $n\geq 4$ people, if every $3$ people have exactly $1$ common friend, how many different values can $n$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of the above} $

Give a family of curves $2(2\sin\theta-\cos\theta+3)x^2-(8\sin\theta+\cos\theta+1)=0$, where $\theta$ is a parameter. Find the maximum value of the length of the chord that $y=2x$ intersects the curve.

Let $a,b,c\ge0$ and $a+b+c\ge3.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

Let $A$ be a given point outside a given circle. Determine points $B, C, D$ on the circle such that the quadrilateral $ABCD$ is convex and has the maximum area .

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

A sequence of integers $a_{1}, a_{2}, a_{3}, \cdots$ is defined as follows: $a_{1}=1$, and for $n \ge 1$, $a_{n+1}$ is the smallest integer greater than $a_{n}$ such that $a_{i}+a_{j} \neq 3a_{k}$ for any $i, j, $ and $k$ in $\{1, 2, 3, \cdots, n+1 \}$, not necessarily distinct. Determine $a_{1998}$.

Albert writes $2025$ numbers $a_1, \ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1].$ Then, each second, he [i]simultaneously[/i] replaces $a_i$ with $\max(a_{i-1},a_i,a_{i+1})$ for all $i = 1, 2, \ldots, 2025$ (where $a_0 = a_{2025}$ and $a_{2026} = a_1$). Compute the expected value of the number of distinct values remaining after $100$ seconds.

There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.

[b]p1.[/b] The oil pipeline passes by three villages $A$, $B$, $C$. In the first village, $30\%$ of the initial amount of oil is drained, in the second - $40\%$ of the amount that will reach village $B$, and in the third - $50\%$ of the amount that will reach village $C$ What percentage of the initial amount of oil reaches the end of the pipeline? [b]p2.[/b] There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than $1$). The product of all fractions is equal to $10$. All numerators and denominators are increased by $1$. Can the product of the resulting fractions be greater than $10$? [b]p3.[/b] The garland consists of $10$ light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need $10$ seconds, to screw it in - also $10$ seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb: a) in $10$ minutes, b) in $5$ minutes? [b]p4.[/b] When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every $15$ minutes, and when they run towards each other, they meet once every $5$ minutes. How many times is the speed of a fast runner greater than the speed of a slow runner? [b]p5.[/b] Petya was $35$ minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait $50$ minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts $55$ minutes? [b]p6.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p7.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number 9 mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p8.[/b] They wrote the numbers $1, 2, 3, 4, ..., 1996, 1997$ in a row. Which digits were used more when writing these numbers - ones or twos? How long? [b]p9.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis $in 1996$ jumps if he must not get to points with coordinates divisible by $ 4$ (points $0$, $\pm 4$, $\pm 8$, etc.)? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Find the sum of all positive integers $n$ such that $$\frac{n^3+8n^2+8n+80}{n+7}$$ is an integer. $\text{(A) }31\qquad\text{(B) }57\qquad\text{(C) }66\qquad\text{(D) }87\qquad\text{(E) }112$

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

Prove that there exist an infinite number of odd natural numbers $m_1<m_2<...$ and an infinity of natural numbers $n_1<n_2<...$ ,such that $(m_k,n_k)=1$ and $m_k^4-2n_k^4$ is a perfect square,for all $k\in\mathbb{N}$.

Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true? $\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$ $\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$ $\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$ $\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$ $\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$

Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.