Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?

Alex calculated the value of function $f(n) = n^2 + n + 1$ for each integer from $1$ to $100$. Marina calculated the value of function $g(n) = n^2-n+1$ for the same numbers. Who of them has greater product of values and what is their ratio?

Compute the number of positive integers $n$ less than $10^8$ such that at least two of the last five digits of $$ \lfloor 1000\sqrt{25n^2 + \frac{50}{9}n + 2022}\rfloor$$ are $6$. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^7\right\rfloor \right)$.

How many positive integers are there such that $\left \lfloor \frac m{11} \right \rfloor = \left \lfloor \frac m{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.) $ \textbf{(A)}\ 44 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 52 \qquad\textbf{(D)}\ 54 \qquad\textbf{(E)}\ 56 $

Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below. [asy] size(10cm); pen thickbrown = rgb(0.6, 0.2, 0); pen thickdark = rgb(0.2, 0, 0); pen dashedarrow = linetype("6 6"); pair A = (-1.14, 4.36), B = (-4.46, -1.28), C = (3.32, -2.78); pair D = (-1.479, -1.855), E = (0.727, 1.372), F = (-3.014, 1.176); draw(A--B--C--cycle, thickbrown); draw(A--B, thickdark); draw(B--C, thickdark); draw(C--A, thickdark); draw(D--F, dashedarrow, EndArrow(6)); draw(F--E, dashedarrow, EndArrow(6)); draw(E--D, dashedarrow, EndArrow(6)); dot(A); label("$A$", A, N); dot(B); label("$B$", B, dir(180)); dot(C); label("$C$", C, dir(330)); dot(D); label("$D$", D, S); dot(E); label("$E$", E, NE); dot(F); label("$F$", F, W); [/asy]

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.

Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC. (The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.) [i]Proposed by Ivan Borsenco[/i]

Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.

The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$.

A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?

Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.

The function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. If the equation $f(x) = 0$ has exactly four distinct real roots, then the sum of these roots is A. 0 B. 2 C. 4 D. 6 E. 8

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? ${ \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999 $

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$. Prove that $(a + 1)^6-1$ is also a multiple of $7^k$.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

For a non-negative integer $n$, call a one-variable polynomial $F$ with integer coefficients $n$-[i]good [/i] if: (a) $F(0) = 1$ (b) For every positive integer $c$, $F(c) > 0$, and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$-good for any $n$.

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?