Olympiad problems

2018 CMIMC Number Theory, 9

Tags: euler , function

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\]

2011 Romanian Master of Mathematics, 4

Tags: function , number theory

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

2012 China Second Round Olympiad, 1

Tags: vector , function , geometry

Let $P$ be a point on the graph of the function $y=x+\frac{2}{x}(x>0)$. $PA,PB$ are perpendicular to line $y=x$ and $x=0$, respectively, the feet of perpendicular being $A$ and $B$. Find the value of $\overrightarrow{PA}\cdot \overrightarrow{PB}$.

2013 ELMO Shortlist, 1

Tags: function , algebra

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

2020 Latvia Baltic Way TST, 2

Tags: function , functional equation in r , algebra

Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy equation: $$ f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y) $$ for all reals $x,y$

2022 SG Originals, Q2

Tags: functional equation , set , integer , algebra , function

Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that $$f(A+B)=f(A)+f(B)$$ for all non-empty sets of integers $A$ and $B$. $A+B$ is defined as $\{a+b: a \in A, b \in B\}$.

2022 CIIM, 1

Tags: quadratic , function , integral , calculus

Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.

2011 ISI B.Stat Entrance Exam, 4

Tags: function , integration , inequalities , calculus , calculus computations

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

2003 China Girls Math Olympiad, 5

Tags: inequalities , function , induction , algebra

Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 \equal{} 2,$ and \[ a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.\]

2023 Romania EGMO TST, P2

Tags: function , pigeonhole principle , induction , modular arithmetic , number theory

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

2010 China Girls Math Olympiad, 5

Tags: function , analytic geometry , graphing lines , slope , algebra

Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.

2022 Taiwan TST Round 3, A

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$. [i]Proposed by Ankan Bhattacharya, USA[/i]

2005 AMC 12/AHSME, 24

Tags: algebra , polynomial , quadratic , function

Let $ P(x) \equal{} (x \minus{} 1)(x \minus{} 2)(x \minus{} 3)$. For how many polynomials $ Q(x)$ does there exist a polynomial $ R(x)$ of degree 3 such that $ P(Q(x)) \equal{} P(x) \cdot R(x)$? $ \textbf{(A)}\ 19\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 32$

1990 IMO Shortlist, 25

Tags: function , number theory , algebra , functional equation

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

1990 IMO Longlists, 4

Tags: function , trigonometry , inequalities

Find the minimal value of the function \[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]

1986 National High School Mathematics League, 1

Tags: function , inequalities

Let $-1<a<0$, $\theta=\arcsin a$. Then the solution set to the inequality $\sin x<a$ is $\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}$ $\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}$ $\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}$ $\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}$

2016 Balkan MO Shortlist, A8

Tags: functional equation , functional equation in z , algebra , function , linear function

Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.

2005 SNSB Admission, 1

Tags: function , vectorial spaces , linear algebra , inequalities

[b]a)[/b] Let be three vectorial spaces $ E,F,G, $ where $ F $ has finite dimension, and $ E $ is a subspace of $ F. $ Prove that if the function $ T:F\longrightarrow G $ is linear, then $$ \dim TF -\dim TE\le \dim F-\dim E. $$ [b]b)[/b] Let $ A,B,C $ be matrices of real numbers. Prove that $$ \text{rang} (AB) +\text{rang} (BC) \le \text{rang} (ABC) +\text{rang} (B) . $$

2004 India IMO Training Camp, 3

Tags: trigonometry , function , functional equation , system of equations , algebra

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

1991 Arnold's Trivium, 64

Tags: algebra , function , domain

Does the Cauchy problem $u|_{y=x^2}=1$, $(\nabla u)^2=1$ have a smooth solution in the domain $y\ge x^2$? In the domain $y\le x^2$?

2019 Turkey MO (2nd round), 5

Tags: function , algebra , combinatorics

Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$

1980 Vietnam National Olympiad, 2

Tags: function , inequalities

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

2022-IMOC, N6

Tags: number theory , function

Find all integer coefficient polynomial $P(x)$ such that for all positive integer $x$, we have $$\tau(P(x))\geq\tau(x)$$Where $\tau(n)$ denotes the number of divisors of $n$. Define $\tau(0)=\infty$. Note: you can use this conclusion. For all $\epsilon\geq0$, there exists a positive constant $C_\epsilon$ such that for all positive integer $n$, the $n$th smallest prime is at most $C_\epsilon n^{1+\epsilon}$. [i]Proposed by USJL[/i]

2022 VJIMC, 1

Tags: calculus , integration , inequalities , function

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

2002 AMC 12/AHSME, 17

Tags: function , trigonometry , search , trig identities , cosine double angle

Let $f(x)=\sqrt{\sin^4 x + 4\cos^2 x}-\sqrt{\cos^4x + 4\sin^2x}$. An equivalent form of $f(x)$ is $\textbf{(A) }1-\sqrt2\sin x\qquad\textbf{(B) }-1+\sqrt2\cos x\qquad\textbf{(C) }\cos\dfrac x2-\sin\dfrac x2$ $\textbf{(D) }\cos x-\sin x\qquad\textbf{(E) }\cos2x$