A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.

[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination? [b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$. [b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and $$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.

[b]p1.[/b] Eric is playing Brawl Stars. If he starts playing at $11:10$ AM, and plays for $2$ hours total, then how many minutes past noon does he stop playing? [b]p2.[/b] James is making a mosaic. He takes an equilateral triangle and connects the midpoints of its sides. He then takes the center triangle formed by the midsegments and connects the midpoints of its sides. In total, how many equilateral triangles are in James’ mosaic? [b]p3.[/b] What is the greatest amount of intersections that $3$ circles and $3$ lines can have, given that they all lie on the same plane? [b]p4.[/b] In the faraway land of Arkesia, there are two types of currencies: Silvers and Gold. Each Silver is worth $7$ dollars while each Gold is worth $17$ dollars. In Daniel’s wallet, the total dollar value of the Silvers is $1$ more than that of the Golds. What is the smallest total dollar value of all of the Silvers and Golds in his wallet? [b]p5.[/b] A bishop is placed on a random square of a $8$-by-$8$ chessboard. On average, the bishop is able to move to $s$ other squares on the chessboard. Find $4s$. Note: A bishop is a chess piece that can move diagonally in any direction, as far as it wants. [b]p6.[/b] Andrew has a certain amount of coins. If he distributes them equally across his $9$ friends, he will have $7$ coins left. If he apportions his coins for each of his $15$ classmates, he will have $13$ coins to spare. If he splits the coins into $4$ boxes for safekeeping, he will have $2$ coins left over. What is the minimum number of coins Andrew could have? [b]p7.[/b] A regular polygon $P$ has three times as many sides as another regular polygon $Q$. The interior angle of $P$ is $16^o$ greater than the interior angle of $Q$. Compute how many more diagonals $P$ has compared to $Q$. [b]p8.[/b] In an certain airport, there are three ways to switch between the ground floor and second floor that are 30 meters apart: either stand on an escalator, run on an escalator, or climb the stairs. A family on vacation takes 65 seconds to climb up the stairs. A solo traveller late for their flight takes $25$ seconds to run upwards on the escalator. The amount of time (in seconds) it takes for someone to switch floors by standing on the escalator can be expressed as $\frac{u}{v}$ , where $u$ and $v$ are relatively prime. Find $u + v$. (Assume everyone has the same running speed, and the speed of running on an escalator is the sum of the speeds of riding the escalator and running on the stairs.) [b]p9.[/b] Avanish, being the studious child he is, is taking practice tests to improve his score. Avanish has a $60\%$ chance of passing a practice test. However, whenever Avanish passes a test, he becomes more confident and instead has a $70\%$ chance of passing his next immediate test. If Avanish takes $3$ practice tests in a row, the expected number of practice tests Avanish will pass can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Find $a + b$. [b]p10.[/b] Triangle $\vartriangle ABC$ has sides $AB = 51$, $BC = 119$, and $AC = 136$. Point $C$ is reflected over line $\overline{AB}$ to create point $C'$. Next, point $B$ is reflected over line $\overline{AC'}$ to create point $B'$. If $[B'C'C]$ can be expressed in the form of $a\sqrt{b}$, where $b$ is not divisible by any perfect square besides $1$, find $a + b$. [b]p11[/b]. Define the following infinite sequence $s$: $$s = \left\{\frac{1}{1},\frac{1}{1 + 3},\frac{1}{1 + 3 + 6}, ... ,\frac{1}{1 + 3 + 6 + ...+ t_k},...\right\},$$ where $t_k$ denotes the $k$th triangular number. The sum of the first $2024$ terms of $s$, denoted $S$, can be expressed as $$S = 3 \left(\frac{1}{2}+\frac{1}{a}-\frac{1}{b}\right),$$ where $a$ and $b$ are positive integers. Find the minimal possible value of $a + b$. [b]p12.[/b] Omar writes the numbers from $1$ to $1296$ on a whiteboard and then converts each of them into base $6$. Find the sum of all of the digits written on the whiteboard (in base $10$), including both the base $10$ and base $6$ numbers. [b]p13.[/b] A mountain number is a number in a list that is greater than the number to its left and right. Let $N$ be the amount of lists created from the integers $1$ - $100$ such that each list only has one mountain number. $N$ can be expressed as $$N = 2^a(2^b - c^2),$$ where $a$, $b$ and $c$ are positive integers and $c$ is not divisible by $2$. Find $a + b+c$. (The numbers at the beginning or end of a list are not considered mountain numbers.)[hide]Original problem was voided because the original format of the answer didn't match the result's format. So I changed it in the wording, in order the problem to be correct[/hide] [b]p14.[/b] A circle $\omega$ with center $O$ has a radius of $25$. Chords $\overline{AB}$ and $\overline{CD}$ are drawn in $\omega$ , intersecting at $X$ such that $\angle BXC = 60^o$ and $AX > BX$. Given that the shortest distance of $O$ with $\overline{AB}$ and $\overline{CD}$ is $7$ and $15$ respectively, the length of $BX$ can be expressed as $x - \frac{y}{\sqrt{z}}$ , where $x$, $y$, and $z$ are positive integers such that $z$ is not divisible by any perfect square. Find $x + y + z.$ [hide]two answers were considered correct according to configuration[/hide] [b]p15.[/b] How many ways are there to split the first $10$ natural numbers into $n$ sets (with $n \ge 1$) such that all the numbers are used and each set has the same average? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made. [i]Proposed by Andrew Wen[/i]

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

We want to split the set of natural numbers from $1$ to $3n$, where $n$ is a natural number, into $n$ mutually disjoint sets $\{x,y,z\}$ of three elements such that always holds: $x + y = 3z$. Is this possible for : a) $n = 5$? b) $n=10$? In both cases, provide either such a split or proof that such a split is not possible.

A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one. Prove that their duty was at least an hour and a half long.

Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions : (a) $ x_{0} \equal{} x_{n} \equal{} 0$, and (b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$. Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100\minus{}a$. Find the largest beautiful number.

Let $n$ be the answer to this problem. Box $B$ initially contains n balls, and Box $A$ contains half as many balls as Box $B$. After $80$ balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\frac{p}{q}$ , where $p$, $q$ are positive integers with gcd$(p, q) = 1$. Find $100p + q$.

There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?

a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$; b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$; c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity! d) Express the value of $\sum (2^{a_1}-1)\ldots (2^{a_k}-1)$ where the sum is over all $2^{n-1}$ ways of choosing $(a_1,a_2,\ldots,a_k)$ such that $a_1+a_2+\ldots +a_k=n$, as a function of some Fibonacci term.

A [i]lattice point[/i] in the Cartesian plane is a point whose coordinates are both integers. A [i]lattice polygon[/i] is a polygon all of whose vertices are lattice points. Let $\Gamma$ be a convex lattice polygon. Prove that $\Gamma$ is contained in a convex lattice polygon $\Omega$ such that the vertices of $\Gamma$ all lie on the boundary of $\Omega$, and exactly one vertex of $\Omega$ is not a vertex of $\Gamma$.

A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero. [i]Proposed by Morteza Saghafian[/i]

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$. [list][*]If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$. [*]If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.[/list] Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps. [i]Proposed by Gerhard Woeginger, Austria[/i]

In some village there are $1000$ inhabitants. Every day, each of them shares the news they learned yesterday with all their friends. It is known that any news becomes known to all residents of the village. Prove that it is possible to select $90$ residents so that if you tell all of them at the same time some news, then in $10$ days it will become known to all residents of the village.

In the game of Ric-Rac-Roe, two players take turns coloring squares of a $3 \times 3$ grid in their color; a player wins if they complete a row or column of their color on their turn. If Alice and Bob play this game, picking an uncolored square uniformly at random on their turn, what is the probability that they tie?

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.

In a test, $201$ students are trying to solve $6$ problems.We know that for each of $5$ first problems, there are at least $140$ students, who can solve it. Moreover, there is exactly $60$ students, who can solve $6^{th}$ problem. Show that there exist $2$ students, such that two of them combined are able to solve all $6$ question. (For example, number $1$ do $1,2,3,4$ and number $2$ do $3,5,6$)