Olympiad problems

2015 District Olympiad, 1

Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $ [b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective. [b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.

2012 Online Math Open Problems, 24

Tags: modular arithmetic , function

Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer. [i]Author: Alex Zhu[/i]

2011 Laurențiu Duican, 2

Tags: set theory , function

Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that: $$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$ [i]Cristinel Mortici[/i]

2016 CCA Math Bonanza, L5.3

Tags: function

Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$, $O(x)=|x|\log |x|\log\log |x|$, $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$, and $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$ for any real number $x$ such that the functions are defined. Determine $$C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))).$$ [i]2016 CCA Math Bonanza Lightning #5.3[/i]

2022 Federal Competition For Advanced Students, P2, 1

Tags: divide , number theory , function

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

2007 Junior Balkan MO, 1

Tags: quadratic , inequalities , algebra , function

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2001 China Team Selection Test, 3

Tags: minimization , calculus , polynomial , function , algebra

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

2003 China Team Selection Test, 3

Tags: function , number theory

Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.

1984 IMO Longlists, 52

Tags: trigonometry , geometry , function

Construct a scalene triangle such that \[a(\tan B - \tan C) = b(\tan A - \tan C)\]

1996 South africa National Olympiad, 2

Tags: algebra , trigonometry , inequalities , function

Find all real numbers for which $3^x+4^x=5^x$.

2023 USA IMO Team Selection Test, 6

Tags: function , algebra

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

2009 Putnam, B5

Tags: domain , limit , integration , geometry , function , algebra

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

2003 India IMO Training Camp, 3

Tags: algebra , function

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

1972 IMO Longlists, 40

Tags: calculus , inequalities , trigonometry , function

Prove the inequalities \[\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}\]

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 Switzerland Team Selection Test, 6

Tags: algebra , function , functional equation , polynomial

Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$

2007 Canada National Olympiad, 3

Tags: functional inequalities , polynomial , algebra , function

Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

2020 March Advanced Contest, 1

Tags: number theory , modular arithmetic , function

In terms of $a$, $b$, and a prime $p$, find an expression which gives the number of $x \in \{0, 1, \ldots, p-1\}$ such that the remainder of $ax$ upon division by $p$ is less than the remainder of $bx$ upon division by $p$.

2010 India IMO Training Camp, 11

Tags: function , algebra

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

2004 Germany Team Selection Test, 3

Tags: counting , function , combinatorics , modular arithmetic , decimal representation

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

2009 Today's Calculation Of Integral, 412

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

Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$. (1) Find $ I_0,\ I_{\minus{}1},\ I_2$. (2) Find $ I_1$. (3) Express $ I_{n\plus{}2}$ in terms of $ I_n$. (4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$. (5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results. You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.

1992 National High School Mathematics League, 6

Tags: function

$f(x)$ is a function defined on $\mathbb{R}$, satisfying: $f(10+x)=f(10-x),f(20-x)=-f(20+x)$. Then, $f(x)$ is $\text{(A)}$even function, but not periodic function $\text{(B)}$even function, and periodic function $\text{(C)}$odd function, but not periodic function $\text{(D)}$odd function, and periodic function

1991 Vietnam National Olympiad, 1

Tags: function , algebra

Find all functions $f: \mathbb{R}\to\mathbb{R}$ satisfying: $\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}$ for all $x,y,z \in \mathbb{R}$

2006 Moldova National Olympiad, 10.5

Tags: derivative , function , algebra , inequalities , calculus , logarithm

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

2003 Pan African, 3

Tags: function

Find all functions $f: R\to R$ such that: \[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]