Olympiad problems

2005 Taiwan National Olympiad, 3

$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.

2019 Korea Winter Program Practice Test, 1

Tags: incenter , algebra , functional equation , geometry , function

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

1995 VJIMC, Problem 3

Tags: real analysis , function , continuous

Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?

2001 AIME Problems, 8

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

A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x)=f(2001)$.

2010 Today's Calculation Of Integral, 638

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

Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$ 1978 Chuo university entrance exam/Science and Technology

2010 ISI B.Stat Entrance Exam, 5

Tags: algebra , function

Let $A$ be the set of all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(xy)=xf(y)$ for all $x,y \in \mathbb{R}$. (a) If $f \in A$ then show that $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$ (b) For $g,h \in A$, define a function $g\circ h$ by $(g \circ h)(x)=g(h(x))$ for $x \in \mathbb{R}$. Prove that $g \circ h$ is in $A$ and is equal to $h \circ g$.

2013 Romania National Olympiad, 3

Tags: function , real analysis , integration

Given $a\in (0,1)$ and $C$ the set of increasing functions $f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine: $(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$ $(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$

2017 Korea - Final Round, 3

Tags: function , algebra

For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions. 1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$. 2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$. Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following. For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.

2015 AMC 12/AHSME, 8

Tags: ratio , rectangle , pythagorean theorem , function , geometry

The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? $\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$

1992 IMO Longlists, 8

Tags: functional equation , induction , function , algebra

Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation \[f(f(x)) + af(x) = b(a + b)x.\] Prove that there exists a unique solution of this equation.

2014 ELMO Shortlist, 6

Tags: combinatorics , function

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

2009 Moldova Team Selection Test, 2

Tags: calculus , derivative , inequalities , rearrangement inequality , function , logarithm

[color=darkred]Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i \equal{} \overline{1,n}$, such that $ \sum a_i \equal{} 1$. Prove that $ \small{\dfrac{a_1^{2 \minus{} m} \plus{} a_2 \plus{} ... \plus{} a_{n \minus{} 1}}{1 \minus{} a_1} \plus{} \dfrac{a_2^{2 \minus{} m} \plus{} a_3 \plus{} ... \plus{} a_n}{1 \minus{} a_1} \plus{} ... \plus{} \dfrac{a_n^{2 \minus{} m} \plus{} a_1 \plus{} ... \plus{} a_{n \minus{} 2}}{1 \minus{} a_1}\ge n \plus{} \dfrac{n^m \minus{} n}{n \minus{} 1}}$[/color]

2014 USAJMO, 4

Tags: inequalities , function , pigeonhole principle

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

2010 Today's Calculation Of Integral, 565

Tags: calculus , derivative , symmetry , function , calculus computations , integration

Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.

2004 APMO, 4

Tags: number theory , function , floor function , relatively prime , modular arithmetic

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

2010 Today's Calculation Of Integral, 522

Tags: function , integration , limit , geometry , calculus , derivative , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2003 Gheorghe Vranceanu, 4

Tags: surjectivity , equivalence relation , combinatorics , function

Find the number of functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that $ (f\circ f\circ f)(n)=n+3, $ for any natural numbers $ n. $

2008 AIME Problems, 9

Tags: number theory , polynomial , algebra , relatively prime , function , geometric series , probability

Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.

1998 ITAMO, 6

Tags: algebra , increasing function , multiplicative function , function

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$. (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$. (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$. (c) Does (b) remain true if the word ”completely” is omitted?

1973 Putnam, B5

Tags: function , algebra , quadratic , rational function

(a) Let $z$ be a solution of the quadratic equation $$az^2 +bz+c=0$$ and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$ (b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$

1976 IMO Longlists, 6

Tags: inequalities , function

For each point $X$ of a given polytope, denote by $f(X)$ the sum of the distances of the point $X$ from all the planes of the faces of the polytope. Prove that if $f$ attains its maximum at an interior point of the polytope, then $f$ is constant.

1991 Balkan MO, 4

Tags: algebra , function

Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.

1989 IMO Longlists, 29

Tags: algebra , functional equation , function , complex number

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

2003 USAMO, 5

Tags: calculus , function , inequalities

Let $ a$, $ b$, $ c$ be positive real numbers. Prove that \[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8. \]

1985 Traian Lălescu, 2.1

Tags: real analysis , function , integral , lagrange

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$