Found problems: 5
2021 Korea Junior Math Olympiad, 1
For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions.
[list]
[*] $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$
[*] $x_1+x_2+ \cdots +x_k = n$
[*] $x_1-x_k \le r$
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For all positive integers $s, t \ge 2$, prove that $$A(st, s, t) = A(s(t-1), s, t) = A((s-1)t, s, t).$$
2013 Baltic Way, 6
Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that
\[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\]
Prove that Santa Claus can give each child a gift that this child likes.
2022 Polish Junior Math Olympiad Second Round, 2.
Let $n\geq 1$ be an integer and let $a$ and $b$ be its positive divisors satisfying $a+b+ab=n$. Prove that $a=b$.
2018 USA TSTST, 2
In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it.
We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of $2$ greater than $1$ (i.e.\ of the form $2^n$ for some integer $n \ge 1$).
[i]Victor Wang[/i]
2024 Israel National Olympiad (Gillis), P3
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diagonal[/b] if the two cells in it [i]aren't[/i] in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image.