There are $30$ men with their $30$ wives sitting at a round table. Show that there always exist two men who are on the same distance from their wives. (The seats are arranged at vertices of a regular polygon.)

Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$? $ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad \textbf{(B)}\ x \plus{} 80 \qquad \textbf{(C)}\ \minus{}x \plus{} 118 \qquad \textbf{(D)}\ x \plus{} 118 \qquad \textbf{(E)}\ 0$

Find a relation between the angles of a triangle such that this could be separated in two isosceles triangles by a line. [i]Dan Brânzei[/i]

The coordinate of an object is given as a function of time by $x = 8t - 3t^2$, where $x$ is in meters and $t$ is in seconds. Its average velocity over the interval from $ t = 1$ to $t = 2 \text{ s}$ is $ \textbf{(A)}\ -2\text{ m/s}\qquad\textbf{(B)}\ -1\text{ m/s}\qquad\textbf{(C)}\ -0.5\text{ m/s}\qquad\textbf{(D)}\ 0.5\text{ m/s}\qquad\textbf{(E)}\ 1\text{ m/s} $

A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\triangle$s in a line. [asy] //diagram by kante314 size(3.3cm); defaultpen(linewidth(1)); real r = 0.37; path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle; draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,1)--(3,1)--(3,2)--(0,2)--cycle); draw((1,0)--(1,3)--(2,3)--(2,0)--cycle); draw(circle((3/2,5/2),1/3)); draw(circle((5/2,1/2),1/3)); draw(circle((3/2,3/2),1/3)); draw(shift(0.5,0.38) * equi); draw(shift(1.5,0.38) * equi); draw(shift(0.5,1.38) * equi); draw(shift(2.5,1.38) * equi); draw(shift(0.5,2.38) * equi); draw(shift(2.5,2.38) * equi); [/asy] How many configurations will have three $\triangle$s in a line and three $\bigcirc$s in a line? $\textbf{(A) } 39 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 78 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 96$

[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions: $$ax + 2y = 1$$ $$2x + ay = 1$$ [b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$. [b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$. [b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer? [b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces. PS. You should use hide for answers.

This morning, Emi dropped the watch and from there it started to move more slowly. When, according to the clock, $2$ minutes have passed, in reality it has already been $3$. Now it is $6:25$ pm and the clock says it is $3:30$ pm. What time did Emi drop the watch?

Given a circle $(O)$ with center $O$ and $P$ a point outside $(O)$. $A$ and $B$ are points on $(O)$ such that $PA$ and $PB$ are tangents to $(O)$. The line $\ell$ through $P$ intersects $(O)$ at points $C$ and $D$, respectively ($C$ lies between $P$ and $D$). Line $BF$ is parallel to line $PA$ and intersects line $AC$ and line $AD$ at $E$ and $F$, respectively. Prove that $BE = BF$.

Suppose \(40\) objects are placed along a circle at equal distances. In how many ways can \(3\) objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.

Let $p(x)$ be a polynomial of degree $2022$ such that: $$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$ Find $p(2023)$.

Triangles $MA_2B_2$ and $MA_1B_1$ are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines $A_1A_2$ , $B_1B_2$ have a common point.

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Let $f(x) = \frac{\sin(x)}{x}$ Find $$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.$$

Solve the equation \[ \sqrt {x \plus{} \sqrt {4x \plus{} \sqrt {16x \plus{} \sqrt {\dotsc \plus{} \sqrt {4^{2008}x \plus{} 3}}}}} \minus{} \sqrt {x} \equal{} 1. \] Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.

An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference $\frac56$: $$\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92.$$ The sequence of seven natural numbers $60$, $70$, $84$, $105$, $140$, $210$, $420$ has the property that the sequence inverted numbers (i.e. the row $\frac{1}{60}$, $\frac{1}{70}$, $\frac{1}{84}$, $\frac{1}{105}$, $\frac{1}{140}$, $\frac{1}{210}$,$\frac{1}{420}$ ) is an arithmetic sequence. (a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence? (b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.

Evaluate the sum $$\sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$$

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Alice and Bob play a number game. Starting with a positive integer $n$ they take turns changing the number with Alice going first. Each player may change the current number $k$ to either $k-1$ or $\lceil k/2\rceil$. The person who changes $1$ to $0$ wins. Determine all $n$ such that Alice has a winning strategy.

Compute the number of functions $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$ such that, for all $x \in \{1, \dots, 15\}$, \[ \frac{f(f(x)) - 2f(x) + x}{15} \]is an integer. [i]Proposed by Ankan Bhattacharya[/i]

The minimum of $ \sin \frac{A}{2} \minus{} \sqrt3 \cos \frac{A}{2}$ is attained when $ A$ is $ \textbf{(A)}\ \minus{}180^{\circ} \qquad \textbf{(B)}\ 60^{\circ} \qquad \textbf{(C)}\ 120^{\circ} \qquad \textbf{(D)}\ 0^{\circ} \qquad \textbf{(E)}\ \text{none of these}$

Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$. (1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$. (2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$. (3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.