Olympiad problems

1996 Romania National Olympiad, 1

Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: circles , geometry , collinear , perpendicular

A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.

2016 Latvia Baltic Way TST, 5

Tags: algebra , inequalities

Given real positive numbers $a, b, c$ and $d$, for which the equalities $a^2 + ab + b^2 = 3c^2$ and $a^3 + a^2b + ab^2 + b^3 = 4d^3$ are fulfilled. Prove that $$a + b + d \le 3c.$$

1990 IMO Longlists, 51

Tags: induction , combinatorics , partition , set systems

Determine for which positive integers $ k$ the set \[ X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\}\] can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$

2004 Silk Road, 1

Tags: algebra

Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.

2012 Middle European Mathematical Olympiad, 8

Tags: number theory

For any positive integer $n $ let $ d(n) $ denote the number of positive divisors of $ n $. Do there exist positive integers $ a $ and $b $, such that $ d(a)=d(b)$ and $ d(a^2 ) = d(b^2 ) $, but $ d(a^3 ) \ne d(b^3 ) $ ?

2024 Mexico National Olympiad, 2

Tags: number theory , divisor

Determine all pairs $(a, b)$ of integers that satisfy both: 1. $5 \leq b < a$ 2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order. [b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.

1996 Turkey Team Selection Test, 3

Tags: inequalities

If $0=x_{1}<x_{2}<...<x_{2n+1}=1$ are real numbers with $x_{i+1}-x_{i} \leq h$ for $1 \leq i \leq 2n$, show that $\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}$

2016 Kazakhstan National Olympiad, 4

Tags: altitude , geometry

In isosceles triangle $ABC$($CA=CB$),$CH$ is altitude and $M$ is midpoint of $BH$.Let $K$ be the foot of the perpendicular from $H$ to $AC$ and $L=BK \cap CM$ .Let the perpendicular drawn from $B$ to $BC$ intersects with $HL$ at $N$.Prove that $\angle ACB=2 \angle BCN$.(M. Kunhozhyn)

2008 Croatia Team Selection Test, 1

Tags: inequalities

Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of: $ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$ $ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$

1998 National Olympiad First Round, 12

Tags: ratio , geometry , perimeter , trigonometry

In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval? $\textbf{(A)}\ \frac{2\sqrt{2} \plus{}1}{4} \qquad\textbf{(B)}\ \frac{\sqrt{2} \plus{}1}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad\textbf{(D)}\ \sqrt{2} \minus{}1 \qquad\textbf{(E)}\ \frac{\sqrt{2} \minus{}1}{2}$

2012 AMC 12/AHSME, 20

Tags: geometry , trapezoid , parallelogram , area of a triangle , heron's formula

A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to \[r_1 + r_2 + r_3 + n_1 + n_2?\] $ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $

2012 USA TSTST, 5

Tags: pigeonhole principle , modular arithmetic , number theory , relatively prime , cigarettes

A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties: (a) $x_0=x$; (b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$; (c) $x_n$ is an integer for some $n$.

2021 Bundeswettbewerb Mathematik, 1

Tags: 3d geometry , inequalities

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Revenge ELMO 2023, 3

Tags: algebra

Find all functions $f\colon\mathbb R^+\to\mathbb R^+$ such that \[(f(x)+f(y)+f(z))(xf(y)+yf(z)+zf(x))>(f(x)+y)(f(y)+z)(f(z)+x)\] for all $x,y,z\in\mathbb R^+$. [i]Alexander Wang[/i] [size=59](oops)[/size]

1994 National High School Mathematics League, 2

Tags: complex number

Give two statements: (1) $a,b,c$ are complex numbers, if $a^2+b^2>c^2$, then $a^2+b^2-c^2>0$. (2) $a,b,c$ are complex numbers, if $a^2+b^2-c^2>0$, then $a^2+b^2>c^2$. Then, which is true? $\text{(A)}$ (1) is correct, (2) is correct as well $\text{(B)}$ (1) is correct, (2) is incorrect $\text{(C)}$ (1) is incorrect, (2) is incorrect as well $\text{(D)}$ (1) is incorrect, (2) is correct

2011 AMC 8, 5

Tags:

What time was it $2011$ minutes after midnight on January 1, 2011? $\textbf{(A)} \text{January 1 at 9:31PM}$ $\textbf{(B)} \text{January 1 at 11:51PM}$ $\textbf{(C)} \text{January 2 at 3:11AM}$ $\textbf{(D)} \text{January 2 at 9:31AM}$ $\textbf{(E)} \text{January 2 at 6:01PM}$

1990 Romania Team Selection Test, 3

Tags: number theory , least common multiple , combinatorics

Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number. [i]Laurentiu Panaitopol[/i]

1981 Polish MO Finals, 1

Tags: 3d geometry , geometry

Two intersecting lines $a$ and $b$ are given in a plane. Consider all pairs of orthogonal planes $\alpha$, $\beta$ such that $a \subset \alpha$ and $b\subset \beta$. Prove that there is a circle such that every its point lies on the line $\alpha \cap \beta$ for some $\alpha$ and $\beta$.

2017 Turkey Team Selection Test, 1

Tags: number theory

$m, n $ are positive integers and $p$ is a prime number. Find all triples $(m, n, p)$ satisfying $(m^3+n)(n^3+m)=p^3$

STEMS 2021-22 Math Cat A-B, A2 B4 C1

Tags: number theory

If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.

2001 Tournament Of Towns, 3

Tags: combinatorics

On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?

2009 Mathcenter Contest, 3

Tags: point , combinatorics , combinatorial geometry

Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

CNCM Online Round 2, 1

Tags:

Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$. Proposed by Minseok Eli Park (wolfpack)

1990 India National Olympiad, 7

Tags: trigonometry , geometry

Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let $ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively. Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles. For which position of $ P$ will the triangle $ DEF$ become equilateral?