Olympiad problems

2025 ISI Entrance UGB, 3

Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.

1956 Putnam, A4

Tags: polynomial , calculus , derivative

Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that $f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$

2008 Moldova National Olympiad, 12.6

Tags: calculus , integration , limit , real analysis

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.

2010 N.N. Mihăileanu Individual, 1

Tags: integration , calculus , inequalities , lipschitz , integral inequalities , definite integral , real analysis

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

2009 Princeton University Math Competition, 7

Tags: trigonometry , inequalities , calculus , derivative

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

2019 Simon Marais Mathematical Competition, B1

Tags: calculus , integration

Determine all pairs $(a,b)$ of real numbers with $a\leqslant b$ that maximise the integral $$\int_a^b e^{\cos (x)}(380-x-x^2) \mathrm{d} x.$$

2016 Romania National Olympiad, 3

Tags: function , calculus , integration , inequalities

Let be a real number $ a, $ and a nondecreasing function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Prove that $ f $ is continuous in $ a $ if and only if there exists a sequence $ \left( a_n \right)_{n\ge 1} $ of real positive numbers such that $$ \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , $$ for all natural numbers $ n. $ [i]Dan Marinescu[/i]

1991 Arnold's Trivium, 81

Tags: function , integration , abstract algebra , calculus

Find the Green's function of the operator $d^2/dx^2-1$ and solve the equation \[\int_{-\infty}^{+\infty}e^{-|x-y|}u(y)dy=e^{-x^2}\]

2010 Today's Calculation Of Integral, 553

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

Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.

2009 Today's Calculation Of Integral, 492

Tags: calculus , integration , analytic geometry , parabola , calculus computations , conic

Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis. You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.

2014 ISI Entrance Examination, 7

Tags: calculus , integration , inequalities , rearrangement inequality , real analysis , integral inequality

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}

Today's calculation of integrals, 898

Tags: calculus , integration , calculus computations , logarithm

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

Today's calculation of integrals, 854

Tags: calculus , integration , analytic geometry , geometry , limit , ellipse , conic

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

PEN P Problems, 12

Tags: function , floor function , calculus , integration , logarithm , inequalities

The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

1982 Vietnam National Olympiad, 1

Tags: polynomial , calculus , integration , trigonometry , vieta , algebra

Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$

2000 Romania National Olympiad, 1

Tags: limit , real analysis , continuity , calculus , integration , integral

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

2016 BMT Spring, 6

Tags: algebra , calculus

Amy is traveling on the $xy$-plane in a spaceship where her motion is described by the following equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant $3$ mph , the magnitude of her velocity as she approaches $(1,-1)$ can be expressed as $\sqrt{\frac{a + be^4}{ c + de^2}}$ . Find $\frac{ac}{bd}$ . (You may assume that the initial conditions do allow her to approach $(1,-1)$)

2013 Today's Calculation Of Integral, 899

Tags: calculus , integration , limit , calculus computations

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2005 Gheorghe Vranceanu, 3

Tags: function , calculus , integration , newton-leibniz , ftc , real analysis , periodic function

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

2009 Today's Calculation Of Integral, 473

Tags: calculus , integration , calculus computations

For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\ r^2 & (r\leq x\leq l \plus{} r)\quad \\ (x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$ Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.

2013 Today's Calculation Of Integral, 869

Tags: calculus , integration , trigonometry , calculus computations

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

2013 Online Math Open Problems, 20

Tags: modular arithmetic , calculus , integration , number theory , prime factorization

A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors. [i]Proposed by Evan Chen[/i]

2011 Today's Calculation Of Integral, 727

Tags: calculus , integration , geometry , limit , calculus computations , logarithm

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

2008 Teodor Topan, 3

Tags: limit , geometry , geometric transformation , calculus , calculus computations , logarithm

Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

2012 Putnam, 5

Tags: function , calculus , logarithm

Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]