Found problems: 85335
2019 Junior Balkan Team Selection Tests - Moldova, 1
Let $n$ be a positive integer. From the set $A=\{1,2,3,...,n\}$ an element is eliminated. What's the smallest possible cardinal of $A$ and the eliminated element, since the arithmetic mean of left elements in $A$ is $\frac{439}{13}$.
1978 All Soviet Union Mathematical Olympiad, 263
Given $n$ nonintersecting segments in the plane. Not a pair of those belong to the same straight line. We want to add several segments, connecting the ends of given ones, to obtain one nonselfintersecting broken line. Is it always possible?
2016 Korea Winter Program Practice Test, 3
Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$.
Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$.
Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.
2009 Ukraine National Mathematical Olympiad, 1
Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$
2019 Taiwan TST Round 1, 2
Given a convex pentagon $ ABCDE. $ Let $ A_1 $ be the intersection of $ BD $ with $ CE $ and define $ B_1, C_1, D_1, E_1 $ similarly, $ A_2 $ be the second intersection of $ \odot (ABD_1),\odot (AEC_1) $ and define $ B_2, C_2, D_2, E_2 $ similarly. Prove that $ AA_2, BB_2, CC_2, DD_2, EE_2 $ are concurrent.
[i]Proposed by Telv Cohl[/i]
2012 Indonesia TST, 4
Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.
1969 IMO Longlists, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
1990 China Team Selection Test, 4
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.
2018 Lusophon Mathematical Olympiad, 4
Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.
1964 Miklós Schweitzer, 1
Among all possible representations of the positive integer $ n$ as $ n\equal{}\sum_{i\equal{}1}^k a_i$ with positive integers $ k, a_1 < a_2 < ...<a_k$, when will the product $ \prod_{i\equal{}1}^k a_i$ be maximum?