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]

In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

Let $ABCD$ be a quadrilateral that has an incircle with centre $O$ and radius $r$. Let $P = AB \cap CD$, $Q = AD \cap BC$, $E = AC \cap BD$. Show that $OE \cdot d = r^2$, where $d$ is the distance of $O$ from $PQ$.

Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds $$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$ Under what conditions does equality occur?

Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$, $BC = 21$, and $CA = 17$. Points $D$ and E lie on side $BC$ such that $BD = 4$, $DE = 6$, and $EC = 11$. The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$, and line $IP$ meets the altitude from $A$ to $BC$ at $X$. Find $(DX \cdot EX)^2$.

Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $n=10^{6}+1$ are there? Proposed by Nikolai Beluhov, Bulgaria

A finite set $K$ consists of at least 3 distinct positive integers. Suppose that $K$ can be partitioned into two nonempty subsets $A,B\in K$ such that $ab+1$ is always a perfect square whenever $a\in A$ and $b\in B$. Prove that \[\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,\]where $|X|$ stands for the cartinality of the set $X$, and for $x\in \mathbb{R}$, $\lfloor x\rfloor$ is the greatest integer that does not exceed $x$.

Through vertices $A$ and $B$ of the unit square $ABCD$ , passes a circle intersecting lines $AD$ and $AC$ at points $K$ and $M$, other than $A$. Find the length of the projection $KM$ onto $AC$.

Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that (a) $M$ is the midpoint of $AB$; (b) $N$ is the midpoint of $AC$; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$. Prove that $\angle APM = \angle PBA$.

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$ [list=a] [*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$. [*] Give an example of function which satisfies the hypothesis.

A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$ (D. Pirshtuk)