Olympiad problems

1986 Traian Lălescu, 1.2

Tags: function , algebra

Prove that there exists a surjective function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that for all natural numbers $ n\ge 2, $ there exists an infinite set $ A_n $ such that $ f(x)=n, $ for all $ x\in A_n. $

1997 India National Olympiad, 6

Tags: function , absolute value , algebra

Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that \[ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}. \]

1999 Miklós Schweitzer, 6

Tags: hilbert space , real analysis , function

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

2009 Harvard-MIT Mathematics Tournament, 5

Tags: function

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

1982 IMO Longlists, 29

Tags: function , limit , algebra

Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.

2003 Purple Comet Problems, 23

Tags: function

For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$, $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$. Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$.

2009 Today's Calculation Of Integral, 514

Tags: calculus , integration , inequalities , trigonometry , function , algebra , domain

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

1989 IMO Shortlist, 10

Tags: function , algebra , functional equation , 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 \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

2007 Junior Balkan MO, 2

Tags: trigonometry , incenter , function , circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2000 Putnam, 4

Tags: calculus , integration , limit , trigonometry , function

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

1984 IMO Longlists, 27

Tags: function , algebra

The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and \[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\] for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.

2007 ISI B.Stat Entrance Exam, 2

Tags: calculus , function , trigonometry , derivative , calculus computations

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

2010 IMC, 5

Tags: function , algebra , polynomial , vector , analytic geometry , real analysis , number theory

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

2024 CCA Math Bonanza, I2

Tags: function

Let $S(x) = x+1$ and $V(x) = x^2-1$. Find the sum of the squares of all real solutions to $S(V(S(V(x)))) = 1$. [i]Individual #2[/i]

1999 Moldova Team Selection Test, 3

Tags: function

The fuction $f(0,\infty)\rightarrow\mathbb{R}$ verifies $f(x)+f(y)=2f(\sqrt{xy}), \forall x,y>0$. Show that for every positive integer $n>2$ the following relation takes place $$f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),$$ for every positive integers $x_1,x_2,\ldots,x_n$.

2023 IMC, 7

Tags: calculus , function

Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that \[f(\xi)+\alpha = f'(\xi)\]

2008 ITest, 39

Tags: function , number theory , relatively prime

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

1994 IberoAmerican, 2

Tags: function , ceiling function , combinatorics

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

2003 AIME Problems, 15

Tags: algebra , polynomial , function

Let \[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

MIPT student olimpiad autumn 2024, 3

Tags: function , real analysis

$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

2012 Romania Team Selection Test, 2

Tags: function , inequalities

Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]

2010 AMC 12/AHSME, 20

Tags: trigonometry , function , inequalities , geometric sequence

A geometric sequence $ (a_n)$ has $ a_1\equal{}\sin{x}, a_2\equal{}\cos{x},$ and $ a_3\equal{}\tan{x}$ for some real number $ x$. For what value of $ n$ does $ a_n\equal{}1\plus{}\cos{x}$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2007 Pre-Preparation Course Examination, 1

Tags: function , number theory

a) Find all multiplicative functions $f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*}$ (i.e. that $\forall x,y\in\mathbb Z_{p}^{*}$, $f(xy)=f(x)f(y)$.) b) How many bijective multiplicative does exist on $\mathbb Z_{p}^{*}$ c) Let $A$ be set of all multiplicative functions on $\mathbb Z_{p}^{*}$, and $VB$ be set of all bijective multiplicative functions on $\mathbb Z_{p}^{*}$. For each $x\in \mathbb Z_{p}^{*}$, calculate the following sums :\[\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)\]

2003 Alexandru Myller, 4

Tags: function , calculus , integration

[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ [i]Mihai Piticari[/i]

2001 VJIMC, Problem 3

Tags: limit , calculus , integration , function

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.