How many diagonals does a regular undecagon ($11$-sided polygon) have?

Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$: \begin{itemize} \item If $a$ divides $b$ and $a \in U$ then $b \in U$. \item If $a,b \in U$ then $\gcd(a,b) \in U$. \item If $a,b \notin U$ then $\operatorname{lcm} (a,b) \notin U$. \end{itemize} How many such pseudo-ultrafilters are there? [i]Proposed by Evan Chen[/i]

For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

Let for natural numbers $a,b,c$ and any natural $n$ we have that $(abc)^n$ divides $ ((a^n-1)(b^n-1)(c^n-1)+1)^3$. Prove that then $a=b=c$.

A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$

Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]

Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$

What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? $\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$

Let $n$ be a positive integer. We call a $n$-tuple $(a_1, . . . , a_n)$ of positive integers [i]nice [/i] if $\bullet$ $gcd (a_1, . . . , a_n) = 1$, and $\bullet$ $a_i|a_{i-1} + a_{i+1}$, for all $i = 1, . . . , n$ (we define $a_0 = a_n$ and $a_{n+1} = a1$ here). Find the maximal possible value of the sum $a_1 +...+ a_n$ if $(a_1, . . . , a_n)$ is a nice $n$-tuple.

Cassandra sets her watch to the correct time at noon. At the actual time of $ \text{1: \!\,00 PM}$, she notices that her watch reads $ \text{12: \!\,57}$ and $ 36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads $ \text{10: \!\,00 PM}$? $ \textbf{(A)}\ \text{10: \!\,22 PM and 24 seconds} \qquad\textbf{(B)}\ \text{10: \!\,24 PM}\qquad\textbf{(C)}\ \text{10: \!\,25 PM}$ $ \textbf{(D)}\ \text{10: \!\,27 PM}\qquad\textbf{(E)}\ \text{10: \!\,30 PM}$

Martin invented the following algorithm. Let two irreducible fractions $ \frac{s_1}{t_1}$ and $ \frac{s_2}{t_2}$ be given as inputs, with the numerators and denominators being positive integers. Divide $ s_1$ and $ s_2$ by their greatest common divisor $ c$ and obtain $ a_1$ and $ a_2$, respectively. Similarily, divide $ t_1$ and $ t_2$ by their greatest common divisor $ d$ and obtain $ b_1$ and $ b_2$, respectively. After that, form a new fraction $ \frac{a_1b_2 \plus{} a_2b_1}{t_1b_2}$, reduce it, and multiply the numerator of the result by $ c$. Martin claims that this algorithm always finds the sum of the original fractions as an irreducible fraction. Is his claim correct?

From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \ldots$. More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins $3$, $17$, and $10$.) What is $p+q$? $\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 57 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 59$

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.

Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.) a) Show that the polyhedron $MDNBP$ is a regular pyramid. b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $

Several football teams participated in a tournament where each team played exactly one game against every other team. Is it possible that there were exactly $2024$ games in total?

Let $a_{1} < a_{2} < a_{3} < \cdots $ be an infinite increasing sequence of positive integers in which the number of prime factors of each term, counting repeated factors, is never more than $1987$. Prove that it is always possible to extract from $A$ an infinite subsequence $b_{1} < b_{2} < b_{3} < \cdots $ such that the greatest common divisor $(b_i, b_j)$ is the same number for every pair of its terms.

Find all integer pairs $(x, y)$ such that $$y^2 = x^3 + 2x^2 + 2x + 1.$$

Consider the function $f:[0,1)\rightarrow[0,1)$ defined by $f(x)=2x-\lfloor 2x\rfloor$, where $\lfloor 2x\rfloor$ is the greatest integer less than or equal to $2x$. Find the sum of all values of $x$ such that $f^{17}(x)=x.$ [i]Proposed by Matthew Weiss

There are $ 1988$ birds in $ 994$ cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?

For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively: $ \textbf{(A)}\ -2, 5\qquad\textbf{(B)}\ 5, 25\qquad\textbf{(C)}\ 10, 20\qquad\textbf{(D)}\ 6, 25\qquad\textbf{(E)}\ 14, 25 $

Solve the equation in integer numbers $$y^3-x^3=91$$

[b]a)[/b] Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $ [b]b)[/b] Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants. [b]c)[/b] Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)