Olympiad problems

2012 ELMO Shortlist, 6

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

2011 Today's Calculation Of Integral, 737

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

Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

2014 PUMaC Number Theory B, 8

Tags: integration , calculus , greatest common divisor , floor function , modular arithmetic , number theory

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

2007 Romania Team Selection Test, 1

Tags: calculus , induction , inequalities , derivative , logarithm

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

1986 National High School Mathematics League, 3

Tags: integration , calculus

In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.

2012 Today's Calculation Of Integral, 848

Tags: trigonometry , integration , calculus , logarithm , calculus computations

Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2014 Online Math Open Problems, 17

Tags: trigonometry , analytic geometry , calculus , derivative , pythagorean theorem , geometry

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

2001 China Western Mathematical Olympiad, 1

Tags: floor function , algebra , function , calculus , quadratic

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

2010 VJIMC, Problem 3

Tags: integration , calculus

Prove that there exist positive constants $c_1$ and $c_2$ with the following properties: a) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$

2009 Today's Calculation Of Integral, 463

Tags: calculus computations , trigonometry , integration , calculus

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.

2021 CMIMC Integration Bee, 7

Tags: integration , calculus

$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$ [i]Proposed by Connor Gordon[/i]

2010 Today's Calculation Of Integral, 568

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

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

1997 AIME Problems, 7

Tags: conic , analytic geometry , graphing lines , slope , calculus

A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0,$ the center of the storm is 110 miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2).$

2017 CMIMC Number Theory, 7

Tags: calculus , derivative , number theory

The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

2021 CMIMC Integration Bee, 8

Tags: integration , calculus

$$\int\left(\frac{x-1}{x^2+1}\right)^2e^x\,dx$$ [i]Proposed by Connor Gordon[/i]

2005 Bundeswettbewerb Mathematik, 2

Tags: number theory , calculus

Let $a$ be such an integer, that $3a$ can be written in the form $x^2 + 2y^2$, with integers $x$ and $y$. Prove that the number $a$ can also be written in this form. [b]Additional problems:[/b] [b]a)[/b] Find a general (necessary and sufficent) criterion for an integer $n$ to be of that form. [b]b)[/b] In how many ways can the integer $n$ be represented in that way?

2000 Tuymaada Olympiad, 3

Tags: calculus , integration , combinatorics

Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$). [asy] unitsize(0.5 cm); for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); } for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } } for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]

2012 Today's Calculation Of Integral, 810

Tags: function , calculus computations , integration , calculus

Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$

2010 Princeton University Math Competition, 6

Tags: algebra , integration , quadratic , quadratic formula , calculus

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

2009 Romania National Olympiad, 1

Tags: calculus , integration , real analysis , inequalities , definite integration , find all functions , function

Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and $$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$

2023 CMIMC Integration Bee, 7

Tags: integration , calculus

\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2008 Alexandru Myller, 1

Tags: integration , squeeze theorem , calculus

$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

2005 Today's Calculation Of Integral, 75

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

A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

2011 Today's Calculation Of Integral, 724

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

Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.

2007 AIME Problems, 8

Tags: calculus , integration , derivative , modular arithmetic , rectangle , geometry , quadratic

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.