Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

There are 6 squares in a row. Each one is labeled with the name of Ana or Beto and with a number from 1 to 6, using each number without repetition. Ana and Beto take turns painting each square according to the order of the numbers on the labels. Whoever paints the square will be the person whose name is on the label. When painting, the person can choose to paint the square either red or blue. Beto wins if at the end there are the same number of blue squares as red squares, and Ana wins otherwise. In how many of all the possible ways of labeling the squares can Beto ensure his victory? The following is an example of a labeling of the labels. [asy] size(12cm); draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle); for (int i=1; i<6; ++i) { draw((i,0)--(i,-1)); } for (int i=1; i<6; ++i) { draw((i,0)--(i,-1.25)); } draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle); for (int i=1; i<7; ++i) { draw((i-0.5,-1)--(i-0.5,-1.25)); } label("Ana", (0.25, -1.125)); label("Beto", (1.25, -1.125)); label("Ana", (2.25, -1.125)); label("Beto", (3.25, -1.125)); label("Ana", (4.25, -1.125)); label("Beto", (5.25, -1.125)); label("1", (0.75, -1.125)); label("3", (1.75, -1.125)); label("5", (2.75, -1.125)); label("2", (3.75, -1.125)); label("4", (4.75, -1.125)); label("6", (5.75, -1.125)); [/asy] First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.

Travis Scott says "FEIN'' every $0.8$ seconds. Find the tens digit of the number of times he says "FEIN'' in $1$ minute.

On a $123 \times 123$ board, each square is painted red or blue according to the following conditions: a) Each square painted red that is not on the edge of the board has exactly $5$ blue squares among its $8$ neighboring squares. b) Each square painted blue that is not on the edge of the board has exactly $4$ red squares among its $8$ neighboring squares. Determine the number of red-painted squares on the board.

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.

Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ . Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.

Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?

The isosceles trapezoid $ABCD$ has perpendicular diagonals. The parallel to the bases through the intersection point of the diagonals intersects the non-parallel sides $[BC]$ and $[AD]$ in the points $P$, respectively $R$. The point $Q$ is symmetric of the point $P$ with respect to the midpoint of the segment $[BC]$. Prove that: a) $QR = AD$, b) $QR \perp AD$.

A student divides an integer $m$ by a positive integer $n$, where $n \le 100$, and claims that $\frac{m}{n}=0.167a_1a_2...$ . Show the student must be wrong.

Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\] for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers. [i]Proposed by Ming Hsiao.[/i]

Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that \[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie? $\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

Given a triangle $ABC$ with its incircle touching sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let the median from $A$ intersects $B_1C_1$ at $M$. Show that $A_1M\perp BC$.

Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer $k$ is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point $A$, can move to any point $P$ such that $|AP| \le k$, as long as $P$ is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let $f(k)$ denote the least number of rounds the Devil takes to win. Prove that $$0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).$$ Note: $a(x) = b(x) (1+o(1))$ if $\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1$.

For positive integers $k,n$ with $k\leq n$, we say that a $k$-tuple $\left(a_1,a_2,\ldots,a_k\right)$ of positive integers is [i]tasty[/i] if [list] [*] there exists a $k$-element subset $S$ of $[n]$ and a bijection $f:[k]\to S$ with $a_x\leq f\left(x\right)$ for each $x\in [k]$, [*] $a_x=a_y$ for some distinct $x,y\in [k]$, and [*] $a_i\leq a_j$ for any $i < j$. [/list] For some positive integer $n$, there are more than $2018$ tasty tuples as $k$ ranges through $2,3,\ldots,n$. Compute the least possible number of tasty tuples there can be. Note: For a positive integer $m$, $[m]$ is taken to denote the set $\left\{1,2,\ldots,m\right\}$. [i]Proposed by Vincent Huang and Tristan Shin[/i]

The center of the circumcircle of $\triangle ABC$ is $O$. The incenter of the triangle is $I$, and the intouch triangle is $A_1B_1C_1$. Let $H_1$ be the orthocenter of $\triangle A_1B_1C_1$. Prove that $O$, $I$, and $H_1$ are collinear.

Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

For each positive integer $n$, define $a_n = n(n + 1)$. Prove that $$n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge n^{a_{3n+2}/a_{3n+1}}$$ .

For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares. [i]Finland, Olli Järviniemi[/i]

It is given a convex hexagon $A_1A_2 \cdots A_6$ such that all its interior angles are same valued (congruent). Denote by $a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.$ $a)$ Prove that holds: $ a_1-a_4=a_2-a_5=a_3-a_6 $ $b)$ Prove that if $a_1,a_2,a_3,...,a_6$ satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.

What is the largest integer whose prime factors add to $14$?

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array. [img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img] [b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$. [img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img] [b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img] [b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ . [b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$. Denote by $M$ the midpoint of the side $AB$, and assume that $\measuredangle CMD=90^{\circ}$. Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$. The line $AK$ meets $BD$ at $P$, and the line $BK$ meets $AC$ at $Q$. Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$. [color=red][Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .][/color]