Olympiad problems

2023 Greece JBMO TST, 3

Tags: algebra , inequalities

Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$ When equality holds?

2014 Contests, 2

Tags: trigonometry , geometry , geometric transformation , reflection , projective geometry , trig identities , law of sines

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

2017 Kosovo National Mathematical Olympiad, 1

Tags: number theory

The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$, $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$. Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$

1958 AMC 12/AHSME, 23

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If, in the expression $ x^2 \minus{} 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount: $ \textbf{(A)}\ {\pm 2ax \plus{} a^2}\qquad \textbf{(B)}\ {2ax \pm a^2}\qquad \textbf{(C)}\ {\pm a^2 \minus{} 3} \qquad \textbf{(D)}\ {(x \plus{} a)^2 \minus{} 3}\qquad\\ \textbf{(E)}\ {(x \minus{} a)^2 \minus{} 3}$

2015 BMT Spring, 13

Tags: combinatorics

On a $2\times 40$ chessboard colored black and white in the standard alternating pattern, $20$ rooks are placed randomly on the black squares. The expected number of white squares with only rooks as neighbors can be expressed as $a/b$, where $a$ and $b$ are coprime positive integers. What is $a + b$? (Two squares are said to be neighbors if they share an edge.)

2015 BMT Spring, 4

Tags: number theory

Determine the greatest integer $N$ such that $N$ is a divisor of $n^{13}-n$ for all integers $n$.

1977 Bundeswettbewerb Mathematik, 1

Tags: number theory

Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?

2005 Estonia Team Selection Test, 4

Tags: polynomial , algebra , real root

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

Kyiv City MO Seniors 2003+ geometry, 2009.10.4

Tags: geometry , rhombus , circumcircle , incenter

In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that: a) quadrilateral $IKCF$ is rhombus; b) the side of this rhombus is $\sqrt {DF \cdot EK}$. (Rozhkova Maria)

2001 Korea - Final Round, 1

Tags: function , modular arithmetic , quadratic , relatively prime , number theory

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

2022 Switzerland Team Selection Test, 1

Tags: number theory , divisibility , number base

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

2017 Hanoi Open Mathematics Competitions, 2

Tags: number theory , diophantine , diophantine equation

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2016 Fall CHMMC, 8

Tags: number theory

For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?

2018 South East Mathematical Olympiad, 7

Tags: combinatorics , graph theory

There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/i] to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

2007 National Olympiad First Round, 32

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We are writing either $0$ or $1$ to unit squares of an $8\times 8 $ chessboard. If the sum of numbers is even vertically, horizontally, or diagonally, what is the greatest possible value of the sum of the all numbers on the board? $ \textbf{(A)}\ 32 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 52 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 64 $

2023 Brazil EGMO Team Selection Test, 3

Tags: inequalities , jensen , square root inequality , square root

Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

2015 India IMO Training Camp, 2

Tags: algebra , polynomial

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

2002 Bosnia Herzegovina Team Selection Test, 1

Tags: inequalities

Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that \[\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab} \ge \frac35\]

2005 Tuymaada Olympiad, 4

Tags: inequalities , trigonometry , 3d geometry , tetrahedron , triangle inequality , geometry

In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle. [i]Proposed by L. Emelyanov[/i]

1966 IMO Longlists, 50

Tags: inequalities , trigonometry , geometry

For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

1953 AMC 12/AHSME, 20

Tags: algebra , polynomial

If $ y\equal{}x\plus{}\frac{1}{x}$, then $ x^4\plus{}x^3\minus{}4x^2\plus{}x\plus{}1\equal{}0$ becomes: $ \textbf{(A)}\ x^2(y^2\plus{}y\minus{}2)\equal{}0 \qquad\textbf{(B)}\ x^2(y^2\plus{}y\minus{}3)\equal{}0\\ \textbf{(C)}\ x^2(y^2\plus{}y\minus{}4)\equal{}0 \qquad\textbf{(D)}\ x^2(y^2\plus{}y\minus{}6)\equal{}0\\ \textbf{(E)}\ \text{none of these}$

2012 Turkmenistan National Math Olympiad, 4

Tags: algebra

Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]

2009 Regional Olympiad of Mexico Center Zone, 5

Tags: perpendicular , geometry

Let $ABC$ be a triangle and let $D$ be the foot of the altitude from $A$. Let points $E$ and $F$ on a line through $D$ such that $AE$ is perpendicular to $BE$, $AF$ is perpendicular to $CF$, where $E$ and $F$ are points other than the point $D$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

2025 Turkey EGMO TST, 4

Tags:

Find all positive integers $n$ such that the number \[ \frac{3 + \sqrt{4n + 9}}{2} \] is the sixth smallest positive divisor of $n$.

2016 Putnam, B2

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Define a positive integer $n$ to be [i]squarish[/i] if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, $2016$ is squarish, because the nearest perfect square to $2016$ is $45^2=2025$ and $2025-2016=9$ is a perfect square. (Of the positive integers between $1$ and $10,$ only $6$ and $7$ are not squarish.) For a positive integer $N,$ let $S(N)$ be the number of squarish integers between $1$ and $N,$ inclusive. Find positive constants $\alpha$ and $\beta$ such that \[\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,\] or show that no such constants exist.