Olympiad problems

1988 All Soviet Union Mathematical Olympiad, 465

Show that there are infinitely many triples of distinct positive integers $a, b, c$ such that each divides the product of the other two and $a + b = c + 1$.

2019 Belarusian National Olympiad, 11.5

Tags: number theory

$n\ge 2$ positive integers are written on the blackboard. A move consists of three steps: 1) choose an arbitrary number $a$ on the blackboard, 2) calculate the least common multiple $N$ of all numbers written on the blackboard, and 3) replace $a$ by $N/a$. Prove that using such moves it is always possible to make all the numbers on the blackboard equal to $1$. [i](A. Naradzetski)[/i]

2023 Kyiv City MO Round 1, Problem 2

Tags: algebra

Positive integers $k$ and $n$ are given such that $3 \le k \le n$.Prove that among any $n$ pairwise distinct real numbers one can choose either $k$ numbers with positive sum, or $k-1$ numbers with negative sum. [i]Proposed by Mykhailo Shtandenko[/i]

2002 AMC 12/AHSME, 5

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For how many positive integers $m$ is \[\dfrac{2002}{m^2-2}\] a positive integer? $\textbf{(A) }\text{one}\qquad\textbf{(B) }\text{two}\qquad\textbf{(C) }\text{three}\qquad\textbf{(D) }\text{four}\qquad\textbf{(E) }\text{more than four}$

2008 Tournament Of Towns, 4

Tags: divisible , number theory , factorial

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2001 Tuymaada Olympiad, 3

Tags: polynomial , function , quadratic , algebra

Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ [i]Proposed by A. Golovanov[/i]

2013 Moldova Team Selection Test, 4

Tags: logarithm , limit , algebra

Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$. $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.

2017 Swedish Mathematical Competition, 3

Tags: geometry , geometric inequality , midpoint , 3d geometry

Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?

2024 Bulgaria National Olympiad, 5

Tags: combinatorics

Let $\mathcal{F}$ be a family of $4$-element subsets of a set of size $5^m$, where $m$ is a fixed positive integer. If the intersection of any two sets in $\mathcal{F}$ does not have size exactly $2$, find the maximal value of $|\mathcal{F}|$.

2018 Iran MO (1st Round), 16

Tags: set , algebra

A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$

LMT Speed Rounds, 2011.5

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The unit of a screw is listed as $0.2$ cents. When a group of screws is sold to a customer, the total cost of the screws is computed with the listed price and then rounded to the nearest cent. If Al has $50$ cents and wishes to only make one purchase, what is the maximum possible number of screws he can buy?

2010 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory

Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$

1979 IMO Shortlist, 4

Tags: 3d geometry , prism , combinatorics , pentagon , geometry

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

2023 BMT, Tie 3

Tags: algebra

Compute the real solution for$ x$ to the equation $$(4^x + 8)^4 - (8^x - 4)^4 = (4 + 8^x + 4^x)^4.$$

2024 ELMO Shortlist, G2

Tags: geometry

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

2017 Harvard-MIT Mathematics Tournament, 6

Tags: combinatorics

[b]R[/b]thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?

2020 LMT Fall, 25

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Consider the equation $x^4-24x^3+210x^2+mx+n=0$. Given that the roots of this equation are nonnegative reals, find the maximum possible value of a root of this equation across all values of $m$ and $n$. [i]Proposed by Andrew Zhao[/i]

2019 Federal Competition For Advanced Students, P2, 6

Tags: divide , min , number theory

Find the smallest possible positive integer n with the following property: For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$. (Gerhard J. Woeginger)

2023-IMOC, G4

Tags: geometry

Given triangle $ABC$. $D$ is a point on $BC$. $AC$ meets $(ABD)$ again at $E$,and $AB$ meets $(ACD)$ again at $F$. $M$ is the midpoint of $EF$. $BC$ meets $(DEF)$ again at $P$. Prove that $\angle BAP = \angle MAC$.

2025 Azerbaijan Junior NMO, 4

Tags: combinatorics

A $3\times3$ square is filled with numbers $1;2;3...;9$.The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation? $\text{a)}$ $24,24,25,25$ $\text{b)}$ $20,23,26,29$

1962 Czech and Slovak Olympiad III A, 4

Tags: geometry , geometric transformation , circles , reflection

Consider a circle $k$ with center $S$ and radius $r$. Let a point $A\neq S$ be given with $SA=d<r$. Consider a light ray emitted at point $A$, reflected at point $B\in k$, further reflected in point $C\in k$, which then passes through the original point $A$. Compute the sinus of convex angle $SAB$ in terms of $d,r$ and discuss conditions of solvability.

LMT Team Rounds 2010-20, A29

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Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart. [i]Proposed by Jeff Lin[/i]

2018 Romania Team Selection Tests, 2

Tags: number theory

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

2007 National Olympiad First Round, 26

Tags: least common multiple , greatest common divisor

Let $c$ be the least common multiple of positive integers $a$ and $b$, and $d$ be the greatest common divisor of $a$ and $b$. How many pairs of positive integers $(a,b)$ are there such that \[ \dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1? \] $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $

2019 Sharygin Geometry Olympiad, 3

Tags: geometry

Let $P$ and $Q$ be isogonal conjugates inside triangle $ABC$. Let $\omega$ be the circumcircle of $ABC$. Let $A_1$ be a point on arc $BC$ of $\omega$ satisfying $\angle BA_1P = \angle CA_1Q$. Points $B_1$ and $C_1$ are defined similarly. Prove that $AA_1$, $BB_1$, $CC_1$ are concurrent.