Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.

Let $Q$ be a unit square in the plane: $Q = [0, 1] \times [0, 1]$. Let $T :Q \longrightarrow Q$ be defined as follows: \[T(x, y) =\begin{cases} (2x, \frac{y}{2}) &\mbox{ if } 0 \le x \le \frac{1}{2};\\(2x - 1, \frac{y}{2}+ \frac{1}{2})&\mbox{ if } \frac{1}{2} < x \le 1.\end{cases}\] Show that for every disk $D \subset Q$ there exists an integer $n > 0$ such that $T^n(D) \cap D \neq \emptyset.$

Find at least one polynomial $P(x)$ of degree 2001 such that $P(x)+P(1- x)=1$ holds for all real numbers $x$.

Let $n$ be a positive integer. There is a board of $(n + 1) \times (n + 1)$ whose squares are numbered in a diagonal pattern, as as the picture shows. Chepito starts from the lower left square, and moving only up or to the right until he reaches the upper right box. During his tour, Chepito writes down the number of each box on the which made a change of direction, and in the end calculates the sum of all the numbers entered. Determine the maximum value of this sum. [img]https://cdn.artofproblemsolving.com/attachments/e/d/f9dc43092a1407d6fe6f1b2c741af015079946.png[/img]

$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between $4$ and $7$ meters? [asy] // Diagram by TheMathGuyd. Found cubic, so graph is perfect. import graph; size(8cm); int i; for(i=1; i<9; i=i+1) { draw((-0.2,2i-1)--(16.2,2i-1), mediumgrey); draw((2i-1,-0.2)--(2i-1,16.2), mediumgrey); draw((-0.2,2i)--(16.2,2i), grey); draw((2i,-0.2)--(2i,16.2), grey); } Label f; f.p=fontsize(6); xaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); yaxis(-0.5,17.8,Ticks(f, 2.0),Arrow()); real f(real x) { return -0.03125 x^(3) + 0.75x^(2) - 5.125 x + 14.5; } draw(graph(f,0,15.225),currentpen+1); real dpt=2; real ts=0.75; transform st=scale(ts); label(rotate(90)*st*"Elevation (meters)",(-dpt,8)); label(st*"Time (seconds)",(8,-dpt)); [/asy] $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

There are $2016$ different points in the plane. Show that between these points at least $45$ different distances occur.

Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions: i) $g(2013)=g(2014) = 0,$ ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$ Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$

For any positive numbers $a,b,c,x,y, z$, prove the inequality $ \frac{a^3}{x}+ \frac{b^3}{y}+ \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.

We say that a number is [i]angelic[/i] if it is greater than $10^{100}$ and all of its digits are elements of $\{1,3,5,7,8\}$. Suppose $P$ is a polynomial with nonnegative integer coefficients such that over all positive integers $n$, if $n$ is angelic, then the decimal representation of $P(s(n))$ contains the decimal representation of $s(P(n))$ as a contiguous substring, where $s(n)$ denotes the sum of digits of $n$. Prove that $P$ is linear and its leading coefficient is $1$ or a power of $10$. [i]Proposed by Grant Yu[/i]

A group of nine math team members like to play Survev.io. They noticed that the number of hours each of them played this week forms an arithmetic progression. The person who played the least played for $1$ hour, while the most played for $9.$ Find the total number of hours all nine group members spent playing Survev.io this week.

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 2$

Let $a,b,c$ three positive distinct real numbers. Evaluate: \[\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx\]

Let $ABCD$ be an isosceles trapezoid such that $AD = BC$, $AB = 3$, and $CD = 8$. Let $E$ be a point in the plane such that $BC = EC$ and $AE \perp EC$. Compute $AE$.

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect its sides at points $A_1$ and $C_1$, and the circumcircle of this triangle is at points $A_0$ and $C_0$, respectively. Lines $A_1C_1$ and $A_0C_0$ intersect at point P. Prove that the segment connecting $P$ to the center of the incircle of triangle $ABC$ is parallel to $AC$.

Martu wants to build a set of cards with the following properties: • Each card has a positive integer on it. • The number on each card is equal to one of $5$ possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same. Determine the fewest number of cards Martu's set can have and give an example for that number.

Is it possible to write $100$ odd numbers on a line such that the sum of any $5$ consecutive numbers is a square number and the sum of any $9$ consecutive numbers is also a square number?

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x ) $$ ( $[a]$ is the biggest integer not exceeding $a$)

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot N$ cm.