Olympiad problems

2013 USAMTS Problems, 3

For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)

1991 Arnold's Trivium, 20

Tags: derivative , calculus , function , parameterization , inequalities

Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.

1987 IMO Shortlist, 22

Tags: algebra , functional equation , function

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

2012 Today's Calculation Of Integral, 841

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

Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

1980 Putnam, A6

Tags: continuous and differentiable , function

Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1 .$ Determine the largest real number $u$ such that $$u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx $$ for all $f$ in $C.$

2013 IMC, 5

Tags: complex number , function

Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime? [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

1990 India National Olympiad, 3

Tags: algebra , function

Let $ f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that (a) $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$; (b) $ 1900 < f(1990) < 2000$, find the possible values that $ f(1990)$ can take. (Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] \equal{} 3$).

1984 Putnam, B4

Tags: calculus , integration , function

Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$, positive on $(0,\infty)$, such that for all $x>0$ the $y$-coordinate of the centroid of the region $$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$is the same as the average value of $g$ on $[0,x]$.

2006 Grigore Moisil Urziceni, 3

Tags: real analysis , function , rolle , lagrange , primitivability , primitive

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

2004 Italy TST, 3

Tags: algorithm , floor function , function , algebra , induction , quadratic

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

2010 Belarus Team Selection Test, 5.3

Tags: algebra , function , functional inequality , inequalities

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

1999 Federal Competition For Advanced Students, Part 2, 3

Tags: probability , floor function , hyperbola , parabola , conic , function

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

2003 Regional Competition For Advanced Students, 1

Tags: function , inequalities

Find the minimum value of the expression $ \frac{a\plus{}1}{a(a\plus{}2)}\plus{}\frac{b\plus{}1}{b(b\plus{}2)}\plus{}\frac{c\plus{}1}{c(c\plus{}2)}$, where $ a,b,c$ are positive real numbers with $ a\plus{}b\plus{}c \le 3$.

PEN P Problems, 3

Tags: floor function , algebra , function , additive number theory

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

2006 IberoAmerican Olympiad For University Students, 4

Tags: real analysis , algebra , integration , calculus , polynomial , function

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2018 Dutch IMO TST, 2

Tags: algebra , inequalities , functional inequality , function

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

1985 Iran MO (2nd round), 5

Tags: algebra , derivative , limit , calculus , function

Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying \[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\] Find the derivative of $f$ in an arbitrary point $x.$

2016 IMO Shortlist, N6

Tags: divisibility , number theory , function

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2010 IberoAmerican Olympiad For University Students, 1

Tags: geometry , function , trigonometry

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

2004 AIME Problems, 15

Tags: function

For all positive integers $ x$, let \[ f(x) \equal{} \begin{cases}1 & \text{if }x \equal{} 1 \\ \frac x{10} & \text{if }x\text{ is divisible by 10} \\ x \plus{} 1 & \text{otherwise}\end{cases}\]and define a sequence as follows: $ x_1 \equal{} x$ and $ x_{n \plus{} 1} \equal{} f(x_n)$ for all positive integers $ n$. Let $ d(x)$ be the smallest $ n$ such that $ x_n \equal{} 1$. (For example, $ d(100) \equal{} 3$ and $ d(87) \equal{} 7$.) Let $ m$ be the number of positive integers $ x$ such that $ d(x) \equal{} 20$. Find the sum of the distinct prime factors of $ m$.

2004 China Girls Math Olympiad, 4

Tags: function , combinatorics

A deck of $ 32$ cards has $ 2$ different jokers each of which is numbered $ 0$. There are $ 10$ red cards numbered $ 1$ through $ 10$ and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered $ k$, then the value of the card is $ 2^k$, and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value $ 2004$.

Today's calculation of integrals, 861

Tags: analytic geometry , calculus , calculus computations , integration , function , geometry

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

2023 Serbia National Math Olympiad, 5

Tags: algebra , calculus , function , functional equation

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.

2017 Romania National Olympiad, 1

Tags: sequence , real analysis , continuity , function

Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.

1999 Irish Math Olympiad, 2

Tags: number theory , algebra , function , prime number

A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies: $ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime; $ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$. Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$