Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

Find the number of ordered triples of integers $(a,b,c)$ such that $$a^2 + b^2 + c^2 - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2$$ and $|a|, |b|, |c| \le 2021.$

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are arranged in a circle. It turned out that for any $i = 1, 2, \ldots, 2024$, the following condition holds: $a_ia_{i+1} < a_{i+2}$. (Here we assume that $a_{2025} = a_1$ and $a_{2026} = a_2$). What largest number of positive integers could there be among these numbers $a_1, a_2, \ldots, a_{2024}$? [i]Proposed by Mykhailo Shtandenko[/i]

Consider $2$ positive integers $a,b$ such that $a+2b=2020$. (a) Determine the largest possible value of the greatest common divisor of $a$ and $b$. (b) Determine the smallest possible value of the least common multiple of $a$ and $b$.

Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/i]

Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.

Suppose P is a point in the interior of a triangle ABC, that x; y; z are the distances from P to A; B; C, respectively, and that p; q; r are the per- pendicular distances from P to the sides BC; CA; AB, respectively. Prove that $xyz \geq 8pqr$; with equality implying that the triangle ABC is equilateral.

[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers? [b]p2.[/b] The following is known about two numbers $x$ and $y$: if $x\ge 0$, then $y = 1 -x$; if $y\le 1$, then $x = 1 + y$; if $x\le 1$, then $x = |1 + y|$. Find $x$ and $y$. [b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method? [b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ . b) Is it possible to come up with $1999$ such triplets? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Positive real numbers $a,b,c,d$ satisfy the equalities \[a^3+b^3+c^3=3d^3\\ b^4+c^4+d^4=3a^4\\ c^5+d^5+a^5=3b^5. \] Prove that $a=b=c=d$.