Let $M$ be a non-empty set of positive integers and let $S_M$ be the sum of all the elements of $M$. We define the [i]tlacoyo[/i] of $M$ as the sum of the digits of $S_M$. For example, if $M=\{2,7,34\}$, then $S_M=2+7+34=43$ and the tlacoyo of the set $M$ is $4+3=7$. \\ Prove that for every positive integer $n$, there exists a set $M$ of $n$ distinct positive integers, such that all its non-empty subsets have the same tlacoyo.

A set of integers is called [b]legendary[/b] if you can reach any integer from it by using the following action multiple times: If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set. Prove that any legendary set contains at least 8 numbers.

Find the largest possible value of the expression \( y - x \), if the non-negative real numbers \( x, y \) satisfy the equation: \[ x^4 = y(y - 2025)^3. \] [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus. [i]Vlad Robu[/i]

Zou and Chou are practicing their 100-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is 3, then the sum of the first two terms of the series is $ \textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 8/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 9/2$

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$

For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.

Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right. [i]L. Kuptsov[/i]

A positive integer is called [i]uphill[/i] if the digits in its decimal representation form a non-decreasing sequence from left to right. That is, a number with decimal representation $\overline{a_1a_2\cdots{}a_d}$ is uphill if $a_i\leq{}a_{i+1}$ for all $i$ (All single-digit integers are uphill.) Given a positive integer $n$, let $f(n)$ be the smallest nonnegative integer $m$ such that $n+m$ is uphill. For example, $f(520)=35$ and $f(169)=0$. Find, with proof, the value of$$f(1)-f(2)+f(3)-f(4)+\cdots{}+f(10^{2018}-1)$$

Call a positive whole number [i]rickety [/i] if it is three times the product of its digits. There are two $2$-digit numbers that are rickety. What is their sum?

In the planetary system of the star Zoolander there are $2015$ planets. On each planet an astronomer lives who observes the closest planet into his telescope (the distances between planets are all different). Prove that there is a planet who is observed by nobody.

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\] Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$. [i]Proposed by Untro368.[/i]

Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $ Prove that $ 0=\det\left( A+\text{adj} A \right) . $ [i]Neculai Moraru[/i]

Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$ (RM Kuznec)

[u]Part 1[/u] You might think this round is broken after solving some of these problems, but everything is intentional. [b]1.1.[/b] The number $n$ can be represented uniquely as the sum of $6$ distinct positive integers. Find $n$. [b]1.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = 5$ and $AC^2 = 1188$. Find $AB$. [b]1.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $15 + 5k$ cents. The amount it made in sales for the previous month was $\$40.50$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]1.4.[/b] Let $A$ be the number of dierent ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 1$, $1 \times 3$, and $1 \times 6$. Let B be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 2$ and $1 \times 5$, where there are 2 different colors available for the $1 \times 2$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]1.5.[/b] An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$. [b]1.6.[/b] Let $x$ be a positive integer such that $3$, $ \log_6(12x)$, $\log_6(18x)$ form an arithmetic progression in some order. Find $x$. [u]Part 2[/u] Oops, it looks like there were some [i]intentional [/i] printing errors and some of the numbers from these problems got removed. Any $\blacksquare$ that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together. [b]2.1.[/b] The number $n$ can be represented uniquely as the sum of $\blacksquare$ distinct positive integers. Find $n$. [b]2.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = \blacksquare$ and $AC^2 = 1536$. Find $AB$. [b]2.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a $\$50$ dollar budget to divide between production and advertising. If it spent k dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $25 + 5k$ cents. The amount it made in sales for the previous month was $\$\blacksquare$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]2.4.[/b] Let $A$ be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times \blacksquare$, $1 \times \blacksquare$, and $1 \times \blacksquare$. Let $B$ be the number of different ways to tile a $1\times n$ rectangle with tiles of size $1 \times \blacksquare$ and $1 \times \blacksquare$, where there are $\blacksquare$ different colors available for the $1 \times \blacksquare$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]2.5.[/b] An integer $n \ge \blacksquare$ is such that $n$ when represented in base $9$ is written the same way as $2n$ is in base $\blacksquare$. Find $n$. [b]2.6.[/b] Let $x$ be a positive integer such that $1$, $\log_{96}(6x)$, $\log_{96}(\blacksquare x)$ form an arithmetic progression in some order. Find $x$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

Consider the sequence of numbers: $ 4, 7, 1, 8, 9, 7, 6, \ldots .$ For $ n > 2$, the $ n$th term of the sequence is the units digit of the sum of the two previous terms. Let $ S_n$ denote the sum of the first $ n$ terms of this sequence. The smallest value of $ n$ for which $ S_n > 10,000$ is: $ \textbf{(A)}\ 1992 \qquad \textbf{(B)}\ 1999 \qquad \textbf{(C)}\ 2001 \qquad \textbf{(D)}\ 2002 \qquad \textbf{(E)}\ 2004$

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?