Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell? (a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$ (b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

Let $(x_n)_{n\geq 1}$ be an increasing and unbounded sequence of positive integers such that $x_1=1$ and $x_{n+1}\leq 2x_n$ for all $n\geq 1$. Prove that every positive integer can be written as a finite sum of distinct terms of the sequence. [i]Note:[/i] Two terms $x_i$ and $x_j$ of the sequence are considered distinct if $i\neq j$.

The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.

The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is [asy] draw(circle((0,0),3)); draw(circle((7,0),3)); draw((0,0)--(3,0)); draw((0,-3)--(0,3)); draw((7,3)--(7,0)--(7+3*sqrt(3)/2,-3/2)); draw((7,0)--(7-3*sqrt(3)/2,-3/2)); draw((0,5)--(0,3.5)--(-0.5,4)); draw((0,3.5)--(0.5,4)); draw((7,5)--(7,3.5)--(6.5,4)); draw((7,3.5)--(7.5,4)); label("$3$",(-0.75,0),W); label("$1$",(0.75,0.75),NE); label("$2$",(0.75,-0.75),SE); label("$6$",(6,0.5),NNW); label("$5$",(7,-1),S); label("$4$",(8,0.5),NNE); [/asy] $\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{4}{9}$

Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$

Triangle $ ABC$ has a right angle at $ B$, $ AB \equal{} 1$, and $ BC \equal{} 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$? [asy]unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("$1$",midpoint(A--B),W); label("$B$",B,SW); label("$D$",D,S); label("$C$",C,SE); label("$A$",A,NW); draw(rightanglemark(C,B,A,2));[/asy]$ \textbf{(A)}\ \frac {\sqrt3 \minus{} 1}{2} \qquad \textbf{(B)}\ \frac {\sqrt5 \minus{} 1}{2} \qquad \textbf{(C)}\ \frac {\sqrt5 \plus{} 1}{2} \qquad \textbf{(D)}\ \frac {\sqrt6 \plus{} \sqrt2}{2}$ $ \textbf{(E)}\ 2\sqrt3 \minus{} 1$

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

An ellipse $\Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\Gamma$.

Given a tetrahedron $ABCD$ and inside the tetrahedron points $K, L, M, N$ that do not lie on a plane. Denote also the centroids of $P$, $Q$, $R$, $S$ of the tetrahedrons $KBCD$, $ALCD$, $ABMD$, $ABCN$ do not lie on a plane. Let $T$ be the centroid of the tetrahedron ABCD, $T_o$ be the centroid of the tetrahedron $PQRS$ and $T_1$ be the centroid of the tetrahedron $KLMN$. a) Prove that the points $T, T_0, T_1$ lie in one straight line. b) Determine the ratio $|T_0T| : |T_0 T_1|$.

Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear. [b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\] Where $|XY|$ means the length of line segment between $X$ and $Y$. [b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out. Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\] Time allowed for this problem was 2 hours.

[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$. [b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make? [b]p3.[/b] Let $ABCD$ be a rhombus such that $\vartriangle ABD$ and $\vartriangle BCD$ are equilateral triangles. Find the angle measure of $\angle ACD$ in degrees. [b]p4.[/b] Find the units digit of $2019^{2019}$. [b]p5.[/b] Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same. [b]p6.[/b] Kathy rolls two fair dice numbered from $1$ to $6$. At least one of them comes up as a $4$ or $5$. Compute the probability that the sumof the numbers of the two dice is at least $10$. [b]p7.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $20x +19y = 2019$. [b]p8.[/b] Let $p$ be a prime number such that both $2p -1$ and $10p -1$ are prime numbers. Find the sum of all possible values of $p$. [b]p9.[/b] In a square $ABCD$ with side length $10$, let $E$ be the intersection of $AC$ and $BD$. There is a circle inscribed in triangle $ABE$ with radius $r$ and a circle circumscribed around triangle $ABE$ with radius $R$. Compute $R -r$ . [b]p10.[/b] The fraction $\frac{13}{37 \cdot 77}$ can be written as a repeating decimal $0.a_1a_2...a_{n-1}a_n$ with $n$ digits in its shortest repeating decimal representation. Find $a_1 +a_2 +...+a_{n-1}+a_n$. [b]p11.[/b] Let point $E$ be the midpoint of segment $AB$ of length $12$. Linda the ant is sitting at $A$. If there is a circle $O$ of radius $3$ centered at $E$, compute the length of the shortest path Linda can take from $A$ to $B$ if she can’t cross the circumference of $O$. [b]p12.[/b] Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan. [b]p13.[/b] Find the number of subsets of $\{1, 2, 3, 4, 5, 6,7\}$ which contain four consecutive numbers. [b]p14.[/b] Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes. [b]p15.[/b] There are $10$ distinct subway lines in Boston, each of which consists of a path of stations. Using any $9$ lines, any pair of stations are connected. However, among any $8$ lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through? [b]p16.[/b] There exist positive integers $k$ and $3\nmid m$ for which $$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.$$ Find the value $k$. [b]p17.[/b] Geronimo the giraffe is removing pellets from a box without replacement. There are $5$ red pellets, $10$ blue pellets, and $15$ white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed. [b]p18.[/b] Find the remainder when $$70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)$$ is divided by $71$. [b]p19.[/b] Let $A_1A_2...A_{12}$ be the regular dodecagon. Let $X$ be the intersection of $A_1A_2$ and $A_5A_{11}$. Given that $X A_2 \cdot A_1A_2 = 10$, find the area of dodecagon. [b]p20.[/b] Evaluate the following infinite series: $$\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}$$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 0 $

In a planet $Papella$ year has $400$ days with days coundting from $1-400$ . A holiday would be that day which is divisible by $6$ . The new gonverment decide to reform a new calendar and split in $10$ months with $40$ day each month , and they decide that day of month which is divisible by $6$ to be holiday . Show that after reform the number of holidays after one year decreased less then $ 10 $ percent .

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

The point in the $xy$-plane with coordinates $(1000,2012)$ is reflected across line $y=2000$. What are the coordinates of the reflected point? $ \textbf{(A)}\ (998,2012) \qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) $

$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$. $\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

Given a triangle $ABC$ , a circle centered at some point $O$ meets the segments $BC$ , $CA$ , $AB$ in the pairs of points $X$ and $X^{'}$ , $Y$ and $Y^{'}$ , $Z$ and $Z^{'}$ , respectively ,labelled in circular order : $X,X^{'},Y,Y^{'},Z,Z^{'}$. Let $M$ be the Miquel point of the triangle $XYZ$ and let $M^{'}$ be that of the triangle $X^{'}Y^{'}Z^{'}$ . Prove that the segments $OM$ and $OM^{'}$ have equal lehgths.

Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$. [i]Proposed by Yang Liu[/i]

Find the largest natural number $ k $ which has the property that there is a partition of the natural numbers $ \bigcup_{1\le j\le k} V_j, $ an index $ i\in\{ 1,\ldots ,k \} $ and three natural numbers $ a,b,c\in V_i, $ satisfying $ a+2b=4c. $

On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e $$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$ If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$. [i]K. Petrov[/i]

[i]The Algorithm.[/i] There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane: \[\begin{array}{ccc}A=(1,10)&B=(976,9)&C=(666,87)\\D=(377,422)&E=(535,488)&F=(775,488) \\ G=(941,500) & H=(225,583)&I=(388,696)\\J=(3,713)&K=(504,872)&L=(560,934)\\&M=(22,997)&\end{array}\] At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the [i]total downtime[/i] of the computers? List the points he visits in order. Your score will be $\left\lfloor \dfrac{N}{40}\right\rfloor$, where \[N=1000+\lfloor\text{the optimal downtime}\rfloor - \lfloor \text{your downtime}\rfloor ,\] or $0$, whichever is greater. By total downtime we mean the sum \[\sum_{P\in S}t_P,\] where $t_P$ is the time at which the repairman reaches $P$.

There are $50$ exact watches lying on a table. Prove that there exist a certain moment, when the sum of the distances from the centre of the table to the ends of the minute hands is more than the sum of the distances from the centre of the table to the centres of the watches.