Olympiad problems

2005 Romania National Olympiad, 1

Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

1970 IMO Longlists, 25

Tags: derivative , real analysis , calculus , function , inequalities , integration

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

2022 JHMT HS, 7

Tags: integration , calculus

Let $a$ be the unique real number $x$ satisfying $xe^x = 2$. Find a closed-form expression for \[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \] You may express your answer in terms of elementary operations, functions, and constants.

2024 CMIMC Integration Bee, 6

Tags: integration , calculus

\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x^2-1}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2020 LIMIT Category 2, 20

Tags: countable , convergence , sequence , calculus

Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) $\{a_n \}_n$ is bounded (B) $\{a_n \}_n$ is unbounded (C) The set of convergent subsequence $\{a_n \}_n$ is countable (D) None of these

2009 Moldova National Olympiad, 12.1

Tags: definite integral , cosinus , integration , calculus

Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx$.

2005 South East Mathematical Olympiad, 8

Tags: function , derivative , trigonometry , inequalities , calculus

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

2009 Today's Calculation Of Integral, 464

Tags: logarithm , calculus , integration , calculus computations

Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.

Today's calculation of integrals, 858

Tags: perimeter , calculus computations , calculus , integration , geometry

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

1991 Arnold's Trivium, 50

Tags: calculus , integration

Calculate \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]

2009 Today's Calculation Of Integral, 434

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

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

2011 China Girls Math Olympiad, 3

Tags: conic , calculus , function , hyperbola , inequalities

The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.

2013 Bogdan Stan, 3

Tags: indefinite integral , antiderivative , calculus , integration

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

2002 APMO, 1

Tags: derivative , calculus , floor function , logarithm , inequalities , function , induction

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

2009 Today's Calculation Of Integral, 468

Tags: calculus , function , calculus computations , integration

Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

2003 China Western Mathematical Olympiad, 1

Tags: function , quadratic , calculus , induction , integration , algebra

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2013 Kosovo National Mathematical Olympiad, 2

Tags: calculus , algebra , polynomial , integration

Find all integer $n$ such that $n-5$ divide $n^2+n-27$.

2006 Germany Team Selection Test, 2

Tags: calculus , inequalities , algebra

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

1998 VJIMC, Problem 4-M

Tags: riemann sum , calculus , inequalities , integration

Prove the inequality $$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.

2012 Today's Calculation Of Integral, 819

Tags: limit , calculus computations , calculus , trigonometry , integration

For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$ Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$

2010 Today's Calculation Of Integral, 654

Tags: function , limit , calculus , calculus computations , integration

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

2011 Math Prize For Girls Problems, 18

Tags: relatively prime , quadratic , calculus , integration , polynomial , algebra , number theory

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

2011 Morocco National Olympiad, 1

Tags: inequalities , derivative , calculus

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

2014 Indonesia MO Shortlist, N4

Tags: calculus , integration , number theory

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

2008 India National Olympiad, 4

Tags: calculus , combinatorics , integration , analytic geometry

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.