Olympiad problems

1995 AMC 12/AHSME, 7

Tags: function

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 100$

PEN P Problems, 3

Tags: function , algebra , floor 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 $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

2019 Belarus Team Selection Test, 1.1

Tags: functional equation , algebra , function

Does there exist a function $f:\mathbb N\to\mathbb N$ such that $$ f(f(n+1))=f(f(n))+2^{n-1} $$ for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.) [i](I. Gorodnin)[/i]

2011 Albania National Olympiad, 1

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

[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

2012 Romania National Olympiad, 2

Tags: function , geometry , geometric transformation , algebra , domain , reflection , real analysis

[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]

1948 Putnam, A2

Tags: geometry , 3d geometry , sphere , algebra , rational function , function

Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$)

2025 Romania National Olympiad, 2

Tags: function , calculus , inequalities , integration

Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]

1982 IMO Longlists, 42

Tags: function , graph theory , combinatorics

Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.

1983 Iran MO (2nd round), 6

Tags: function

Suppose that \[f(x)=\{\begin{array}{cc}n,& \qquad n \in \mathbb N , x= \frac 1n\\ \text{} \\x, & \mbox{otherwise}\end{array}\] [b]i)[/b] In which points, the function has a limit? [b]ii)[/b] Prove that there does not exist limit of $f$ in the point $x=0.$

2004 China Western Mathematical Olympiad, 4

Tags: function , euler , search , number theory , totient function , prime number , logarithm

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2015 District Olympiad, 3

Tags: function , continuity , monotone functions , find all functions , real analysis

Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality: $$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$

2004 IberoAmerican, 3

Tags: modular arithmetic , induction , quadratic , function , number theory

Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$

2010 IberoAmerican, 1

Tags: vector , analytic geometry , function , combinatorics

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

2014 Romania Team Selection Test, 4

Tags: function , induction , relatively prime , prime number , number theory

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

2004 Bulgaria National Olympiad, 4

Tags: invariant , induction , function , geometry , geometric transformation , group theory , combinatorics

In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.

2024 Macedonian Mathematical Olympiad, Problem 5

Tags: function , inequalities

Let $f:\mathbb{N} \rightarrow \mathbb{N} \setminus \left \{ 1 \right \}$ be a function which satisfies both the inequality $f(a+f(a)) \leq 2a+3$ and the equation $$f(f(a)+b) = f(a+f(b)),$$ for any two $a,b \in \mathbb{N}$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be defined with: $g(a)$ is the largest prime divisor of $f(a)$. Prove that there exist integers $a>b>2024$ such that $b|a$ and $g(a) = g(b)$.

2008 IMC, 1

Tags: function , algebra , functional equation

Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that \[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]

2010 Slovenia National Olympiad, 3

Tags: function , algebra

Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation \[(y+1)f(x+y) = f\left(xf(y)\right)\] For all non-negative real numbers $x$ and $y.$

1998 Estonia National Olympiad, 3

Tags: function , periodic , functional equation , functional , algebra

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.

2013 AMC 12/AHSME, 25

Tags: function , quadratic , complex number , absolute value

Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? ${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $

2005 Today's Calculation Of Integral, 90

Tags: calculus , integration , limit , factorial , function , ratio , logarithm

Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

1977 AMC 12/AHSME, 11

Tags: function

For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$ (i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true? $\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x$ $\textbf{II. [}x+y\textbf{]}=\textbf{[}x\textbf{]}+\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$ $\textbf{III. [}xy\textbf{]}=\textbf{[}x\textbf{]}\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$ $\textbf{(A) }\text{none}\qquad\textbf{(B) }\textbf{I }\text{only}\qquad\textbf{(C) }\textbf{I}\text{ and }\textbf{II}\text{ only}\qquad\textbf{(D) }\textbf{III }\text{only}\qquad \textbf{(E) }\text{all}$

2011 Today's Calculation Of Integral, 685

Tags: calculus , integration , function , calculus computations

Suppose that a cubic function with respect to $x$, $f(x)=ax^3+bx^2+cx+d$ satisfies all of 3 conditions: \[f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1\]. Find $f(x)$ for which $I=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx$ is minimized, the find the minimum value. [i]2011 Tokyo University entrance exam/Humanities, Problem 1[/i]

2015 Iran MO (3rd round), 2

Tags: function , algebra

Prove that there are no functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x,y\in \mathbb{R}:$ $ f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y$ and $f(x)\geq x^2$. [i]Proposed by Mohammad Ahmadi[/i]

2015 Romania National Olympiad, 3

Tags: function , algebra

Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$ for all $ x,y\in\mathbb{Q} . $