Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?

Find all natural numbers $n < 1978$ with the following property: If $m$ is a natural number, $1 < m < n$, and $(m, n) = 1$ (i.e., $m$ and $n$ are relatively prime), then $m$ is a prime number.

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon. (a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.) (b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter. (c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths. It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle. (d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter. (e) Prove that the polygon of maximum area must be equiangular, or have all angles equal. (f ) Prove that the polygon of maximum area is a regular polygon. PS. You had better use hide for answers.

Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.

Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$. Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac 1{10}$. [i]Yugoslavia[/i]

A geometric progression of positive integers has $n$ terms; the first term is $10^{2015}$ and the last term is an odd positive integer. How many possible values of $n$ are there? [i]Proposed by Evan Chen[/i]

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $

In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$.

Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

Quadrilateral $ABCD$ is inscribed in a circle of radius $6$. If $\angle BDA = 40^o$ and $AD = 6$, what is the measure of $\angle BAD$ in degrees?

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

Lesha put numbers from 1 to $22^2$ into cells of $22\times 22$ board. Can Oleg always choose two cells, adjacent by the side or by vertex, the sum of numbers in which is divisible by 4?

Let $ABCDEF$ be a convex hexagon with sides not parallel and tangent to a circle $\Gamma$ at the midpoints $P$, $Q$, $R$ of the sides AB, $CD$, $EF$ respectively. $\Gamma$ is tangent to $BC$, $DE$ and $FA$ at the points $X, Y, Z$ respectively. Line $AB$ intersects lines $EF$ and $CD$ at points $M$ and $N$ respectively. Lines $MZ$ and $NX$ intersect at point $K$. Let $ r$ be the line joining the center of $\Gamma$ and point $K$. Prove that the intersection of $PY$ and $QZ$ lies on the line $ r$.

There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems. Prove, that they can find problem for contest.

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

Let $n\ge2$ be an integer. A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices. Prove that at least three of these triangles are isosceles.

Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities? [asy] // made by SirCalcsALot size(300); pen shortdashed=linetype(new real[] {6,6}); // axis draw((0,0)--(0,9300), linewidth(1.25)); draw((0,0)--(11550,0), linewidth(1.25)); for (int i = 2000; i < 9000; i = i + 2000) { draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey); label(string(i), (0,i), W); } for (int i = 500; i < 9300; i=i+500) { draw((0,i)--(150,i),linewidth(1.25)); if (i % 2000 == 0) { draw((0,i)--(250,i),linewidth(1.25)); } } int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750}; int data_length = 20; int r = 550; for (int i = 0; i < data_length; ++i) { fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey); draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)); } draw((0,4750)--(11450,4750),shortdashed); label("Cities", (11450*0.5,0), S); label(rotate(90)*"Population", (0,9000*0.5), 10*W); [/asy] $\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the two squares.

A composite positive integer is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite positive integer is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]