Olympiad problems

2010 Today's Calculation Of Integral, 551

Tags: calculus , integration , analytic geometry , geometry , trigonometry , calculus computations

In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.

2011 Putnam, A5

Tags: function , vector , calculus , derivative

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

2014 Contests, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2008 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , calculus computations , logarithm

([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.

2005 Today's Calculation Of Integral, 78

Tags: calculus , integration , trigonometry , calculus computations

Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate \[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

1969 IMO Longlists, 51

Tags: geometry , calculus , counting , function

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

2004 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , limit

Find \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2}-\sqrt[3]{x^3-x^2} \right). \]

2010 Contests, 524

Tags: calculus , integration , function , calculus computations

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2009 Today's Calculation Of Integral, 421

Tags: calculus , integration , limit , calculus computations

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.

2013 Today's Calculation Of Integral, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2001 USA Team Selection Test, 2

Tags: integration , function , calculus , algebra , binomial theorem , combinatorics

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2004 Vietnam National Olympiad, 3

Tags: calculus , integration , quadratic , modular arithmetic , number theory

Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

2013 Today's Calculation Of Integral, 873

Tags: calculus , integration , ellipse , analytic geometry , graphing lines , conic , geometry

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

2012 Today's Calculation Of Integral, 846

Tags: calculus , integration , limit , function , absolute value , calculus computations , logarithm

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

2019 Jozsef Wildt International Math Competition, W. 31

Tags: integration , inequalities , calculus

Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

1982 Putnam, A3

Tags: integration , calculus

Evaluate $$\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.$$

2008 Brazil Team Selection Test, 3

Tags: inequalities , function , algebra , calculus

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2025 Romania EGMO TST, P1

Tags: integration , inequalities , algebra , sequence , summation , calculus

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

1975 Canada National Olympiad, 7

Tags: function , trigonometry , quadratic , calculus , derivative , limit , algebra

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

2012 Pre - Vietnam Mathematical Olympiad, 1

Tags: calculus , derivative , logarithm , inequalities

For $a,b,c>0: \; abc=1$ prove that \[a^3+b^3+c^3+6 \ge (a+b+c)^2\]

2006 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , integration , logarithm

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

2007 Today's Calculation Of Integral, 220

Tags: calculus , integration , inequalities , calculus computations

Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

2010 Today's Calculation Of Integral, 577

Tags: calculus , integration , inequalities , calculus computations

Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

2019 Simon Marais Mathematical Competition, A4

Tags: series , calculus , real analysis , sequence

Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges. Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?