Construct the midpoint of a segment using an unmarked ruler and a [i]trisector[/i] that marks in a segment the two points that divide the segment in three equal parts.

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$. Note that you are not allowed to solve in using partial differentiation here.

There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.

Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.

If $(A, <)$ is a partially ordered set, its dimension, $\dim (A, <)$, is the least cardinal $\kappa$ such that there exist $\kappa$ total orderings $\{ <_{\alpha} \colon \alpha < \kappa \}$ on $A$ with $<=\cap_{\alpha < \kappa} <_\alpha$. Show that if $\dim (A, <)>\aleph_0$, then there exist disjoint $A_0, A_1\subseteq A$ with $\dim (A_0, <)$, $\dim (A_1, <)>\aleph_0$. [D. Kelly, A. Hajnal, B. Weiss]

Let be a natural number $ n, $ denote with $ C $ the square in the complex plane whose vertices are the affixes of $ 2n\pi\left( \pm 1\pm i \right) , $ and consider the set $$ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} $$ Prove the following implications. [b]a)[/b] $ \exists \alpha\in\mathbb{R}_{>0}\quad \forall z\in\partial C\quad \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} $ [b]b)[/b] $ \forall f\in\Lambda\quad\frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} $ [b]c)[/b] $ \forall f\in\Lambda\quad \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4} $

$120$ bags with $100$ coins are placed on the floor. One bag has coins that weigh $9$ grams, the other bags have coins that weigh $10$ grams. One may place some coins (not necessarily from the same bag) on a weighing scale, but it will only properly display the weight if it is less than $1000$ grams. Determine the minimum number of times that the weighing scale may be used in order to identify the bag that has the $9$-gram coins.

Ten women sit in $ 10$ seats in a line. All of the $ 10$ get up and then reseat themselves using all $ 10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated? $ \textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8$

The quadrilateral $ABCD$ is inscribed in the circle $\Gamma$. Let $I_B$ and $I_D$ be the centers of the circles $\omega_B$ and $\omega_D$ inscribed in the triangles $ABC$ and $ADC$, respectively. A common external tangent to $\omega_B$ and $\omega_D$ intersects $\Gamma$ at $K$ and $L{}$. Prove that $I_B,I_D,K$ and $L{}$ lie on the same circle.

Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers: $$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$

Al and Bob are playing Rock Paper Scissors. Al plays rock. What is the probability that Al wins, given that Bob plays randomly and has an equal probability of playing rock, paper, and scissors?

Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$ $$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$ $$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$ $$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$

Let $ABCS$ be an isosceles trapezoid. Denote $A',B',C',D'$ the incenters of triangles $BCD,CDA,$ $DAB,ABC,$ respectively. Show that $A',B',C',D'$ are vertices of a rectangle.

Find the smallest two-digit positive integer that is a divisor of 201020112012.

Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} ($s(n)$ denotes the sum of digits of $n$.) [i]Proposed by Mohammadamin Sharifi[/i]

1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions: [b]I.[/b] The distance between any two points $\in S$ is not greater than 1. [b]II.[/b] For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$. [b]III. [/b] For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$. Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.

Rombus ABCD with acute angle $B$ is given. $O$ is a circumcenter of $ABC$. Point $P$ lies on line $OC$ beyond $C$. $PD$ intersect the line that goes through $O$ and parallel to $AB$ at $Q$. Prove that $\angle AQO=\angle PBC$.

Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$.

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

[b]8.1 [/b] Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals. [b]8.2[/b] It is known that $a,b$ and $\sqrt{a}+\sqrt{b} $ are rational numbers. Prove that then $\sqrt{a}$, $\sqrt{b} $ are rational. [b]8.3 / 9.2[/b] Solve equation $x^3 - [x]=3$ [b]8.4[/b] Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right. . [b]8.5[/b] Given $n$ numbers $x_1, x_2, . . . , x_n$, each of which is equal to $+1$ or $-1$. At the same time $$x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 .$$ Prove that $n$ is divisible by $4$. [b]8.6[/b] There are $n$ points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than $120^0$.Prove that all points lie on an arc of size $120^0$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]