Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.

3. Eight farmers have a checkered $8 \times 8$ field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 8 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 9 berries from the field so that for sure no farmer will notice this? Tatiana Kazitsyna

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

We have a row of 203 cells. Initially the leftmost cell contains 203 tokens, and the rest are empty. On each move we can do one of the following: 1)Take one token, and move it to an adjacent cell (left or right). 2)Take exactly 20 tokens from the same cell, and move them all to an adjacent cell (all left or all right). After 2023 moves each cell contains one token. Prove that there exists a token that moved left at least nine times.

Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same amount of tea at a different constant rate. How many tea crates does Eddie dump per hour? [i]Proposed by Samuel Tsui[/i] [hide=Solution] [i]Solution.[/i] $\boxed{90}$ Sam dumps a total of $6 \cdot 60 = 360$ tea crates and since it takes Eddie $4$ hours to dump that many he dumps at a rate of $\dfrac{360}{4}= \boxed{90}$ tea crates per hour. [/hide]

The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below. [center][img]https://snag.gy/LGimC8.jpg[/img][/center]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite. (It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$). [hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito[/hide]

There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$

Prove, for angles $\alpha$, $\beta$ and $\gamma$ of a right triangle, the following relation: $$\text{sin } \alpha \text{ sin } \beta \text{ sin } (\alpha-\beta) \text{ } + \text{ sin } \beta \text{ sin } \gamma \text{ sin } (\beta-\gamma) \text{ }+ \text{ sin } \gamma \text{ sin } \alpha \text{ sin } (\gamma-\alpha) \text{ }+ \text{ sin } (\alpha-\beta) \text{ sin } (\beta-\gamma) \text{ sin } (\gamma-\alpha) = 0.$$

Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that $AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$ and find for which points $X$ the equality is met.

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

Aisha went shopping. At the first store she spent $40$ percent of her money plus four dollars. At the second store she spent $50$ percent of her remaining money plus $5$ dollars. At the third store she spent $60$ percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping?

A $9\times9$ checkerboard is colored with 2 colors. If we choose any $3\times1$ region on the checkerboard we can paint all of the squares in that region with the color that is in the majority in that region. Show that with a finite number of these operations, we can paint the checkerboard all in the same color.

The positive numbers $a, b$ and $c$ satisfy $a \ge b \ge c$ and $a + b + c \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 \le 1$ . (F . L . Nazarov)

Find all positive integer $n$ for which there exist real number $a_1,a_2,\ldots,a_n$ such that $$\{a_j-a_i|1\le i<j\le n\}=\left\{1,2,\ldots,\frac{n(n-1)}2\right\}.$$

Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. [i] Proposed by Michael Ren [/i]

Given a positive integer $\ell,$ define the sequence $\{a^{(\ell)}\}_{n=1}^{\infty}$ such that $a_n^{(\ell)}=\lfloor n + \sqrt[\ell]{n}+\tfrac{1}{2}\rfloor$ for all positive integers $n.$ Let $S$ denote the set of positive integers that appear in all three of the sequences $\{a_n^{(2)} \}_{n=1}^{\infty},$ $\{a_n^{(3)} \}_{n=1}^{\infty},$ and $\{a_n^{(4)} \}_{n=1}^{\infty}.$ Find the sum of the elements of $S$ that lie in the interval $[1,100].$

Tony plays a game in which he takes $40$ nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself $5$ points for each nickel that lands in the jar, and takes away $2$ points from his score for each nickel that hits the ground. After Tony is done tossing all $40$ nickels, he computes $88$ as his score. Find the greatest number of nickels he could have successfully tossed into the jar.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

A point $ P$ is randomly selected from the rectangular region with vertices $ (0, 0)$, $ (2, 0)$, $ (2, 1)$, $ (0, 1)$. What is the probability that $ P$ is closer to the origin than it is to the point $ (3, 1)$? $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ 1$

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm. (translated by Miklós Maróti)

Let $k$ be a circle with centre $O$. Let $AB$ be a chord of this circle with midpoint $M\neq O$. The tangents of $k$ at the points $A$ and $B$ intersect at $T$. A line goes through $T$ and intersects $k$ in $C$ and $D$ with $CT < DT$ and $BC = BM$. Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.