Olympiad problems

1988 IMO Longlists, 70

Tags: geometry , trigonometry , inradius , inequalities , function , circumcircle

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

2014 Cezar Ivănescu, 2

Tags: quadratic , complex number , trigonometry , geometry , function

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

2013 ELMO Shortlist, 8

Tags: function , inequalities

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

2009 Today's Calculation Of Integral, 483

Tags: calculus computations , limit , logarithm , integration , calculus , function

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

2008 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: function , inequalities

IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1$ prove the inequality: \[ \frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.\]

1987 Yugoslav Team Selection Test, Problem 2

Tags: algebra , function

Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.

2017 Thailand TSTST, 3

Tags: functional equation , function , algebra

Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.

2010 Today's Calculation Of Integral, 611

Tags: derivative , search , logarithm , inequalities , integration , function , calculus

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

2010 India IMO Training Camp, 4

Tags: inequalities , function

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

2015 Turkey Team Selection Test, 7

Tags: sophie germain identity , functional equation , function , algebra

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2012 India IMO Training Camp, 3

Tags: function , induction , inequalities , algebra

Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying \[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\] for all $x, y\in \mathbb{R}^{+}$.

2022 IMO Shortlist, N6

Tags: prime , prime number , number theory , prime factorization , function

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2014 Contests, 3

Tags: function , geometry , algebra , reflection

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

2021 Bangladeshi National Mathematical Olympiad, 7

Tags: perfect cube , divisor , perfect square , function , number theory

For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

2015 AMC 12/AHSME, 18

Tags: function , quadratic , vieta , sfft , polynomial , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2015 NIMO Problems, 7

Tags: function , algebra

Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5. [i]Proposed by Matthew Lerner-Brecher[/i]

2009 China National Olympiad, 1

Tags: algebra , symmetry , polynomial , absolute value , inequalities , function

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

2007 Today's Calculation Of Integral, 234

Tags: integration , trigonometry , calculus computations , function , calculus

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

PEN K Problems, 6

Tags: functional equation , polynomial , function , algebra

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]

2002 IMC, 2

Tags: function , real analysis

Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?

2000 Putnam, 4

Tags: induction , function , trigonometry

Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.

2009 India Regional Mathematical Olympiad, 3

Tags: number theory , function

Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.

2001 India IMO Training Camp, 2

Tags: polynomial , algebra , function

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

2024 Korea Junior Math Olympiad, 8

Tags: function , algebra

$f$ is a function from the set of positive integers to the set of all integers that satisfies the following. [b]$\cdot$[/b] $f(1)=1, f(2)=-1$ [b]$\cdot$[/b] $f(n)+f(n+1)+f(n+2)=f(\left\lfloor\frac{n+2}{3}\right\rfloor)$ Find the number of positive integers $k$ not exceeding $1000$ such that $f(3)+f(6)+\cdots+f(3k-3)+f(3k)=5$.

1985 IMO Longlists, 56

Tags: circumcircle , rhombus , function , geometry

Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$