Olympiad problems

2014 Purple Comet Problems, 27

Tags: probability , vector , algorithm , number theory , relatively prime

Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2020 CHMMC Winter (2020-21), 2

Tags: number theory

Find the sum of all positive integers $x < 241$ such that both $x^{24} + x^{18} + x^{12} + x^6 + 1$ and $x^{20} + x^{10} + 1$ are multiples of $241$.

2024 Thailand TST, 1

Tags: arithmetic sequence , number theory

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

2015 USAMTS Problems, 5

Tags:

Find all positive integers $n$ that have distinct positive divisors $d_1, d_2, \dots, d_k$, where $k>1$, that are in arithmetic progression and $$n=d_1+d_2+\cdots+d_k.$$ Note that $d_1, d_2, \dots, d_k$ do not have to be all the divisors of $n$.

2005 iTest, 2

Tags: algebra

Find the sum of the solutions of $x^3 + x + 182 = 0$.

1992 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , circumradius , circumcircle

Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

2009 USAMTS Problems, 4

Tags: inequalities , pigeonhole principle

Let $S$ be a set of $10$ distinct positive real numbers. Show that there exist $x,y \in S$ such that \[0 < x - y < \frac{(1 + x)(1 + y)}{9}.\]

2016 Israel Team Selection Test, 4

Tags: number theory , greatest common divisor

Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.

Russian TST 2019, P3

Tags: combinatorics

Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. Proposed by [i]India[/i]

2016 AMC 12/AHSME, 23

Tags: probability

Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? $\textbf{(A) }\frac16\qquad\textbf{(B) }\frac13\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac23\qquad\textbf{(E) }\frac56$