Olympiad problems

2005 Harvard-MIT Mathematics Tournament, 6

The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.

2010 Czech-Polish-Slovak Match, 3

Tags: linear algebra , analytic geometry , number theory , matrix , rectangle , calculus , geometry

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

2010 Today's Calculation Of Integral, 624

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

Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

MIPT student olimpiad spring 2024, 1

Tags: integration , calculus

Find integral: $\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$

1997 IMC, 1

Tags: real analysis , function , derivative , calculus

Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$. Define $g(x)$ as follows: \[ \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} \] (a) Show that $g$ is bounded in some neighbourhood of $0$. (b) Is the above true for $f\in C^2(\mathbb{R})$?

2013 Today's Calculation Of Integral, 889

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

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

1966 IMO Shortlist, 30

Tags: logarithm , inequalities , calculus

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

2012 Turkey Junior National Olympiad, 1

Tags: number theory , integration , quadratic , diophantine equation , calculus

Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

1973 Canada National Olympiad, 1

Tags: integration , logarithm , inequalities , calculus

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , logarithm , inequalities , calculus

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

2013 Today's Calculation Of Integral, 874

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

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2014 Contests, 902

Tags: integration , calculus computations , calculus

For $a\geq 0$, find the minimum value of $S(a)=\int_0^1 |x^2+2ax+a^2-1|\ dx.$

2010 Contests, 3

Tags: matrix , geometry , linear algebra , rectangle , calculus , analytic geometry , number theory

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

2009 Today's Calculation Of Integral, 431

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

Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$. (1) Find the maximum and minimum values of $ f(\theta)$. (2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.

2025 ISI Entrance UGB, 3

Tags: isi ugb , calculus

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 $x$.

1953 AMC 12/AHSME, 47

Tags: inequalities , logarithm , calculus

If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2009 Today's Calculation Of Integral, 445

Tags: logarithm , integration , calculus , calculus computations

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2009 Today's Calculation Of Integral, 513

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

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

2007 Today's Calculation Of Integral, 205

Tags: logarithm , integration , calculus , calculus computations

Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

2010 Contests, A3

Tags: derivative , function , complex analysis , vector , calculus , probability

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

2005 Harvard-MIT Mathematics Tournament, 2

Tags: trigonometry , logarithm , calculus , integration

A plane curve is parameterized by $x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u $ and $ y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u $ for $ 1 \le t \le 2 $. What is the length of the curve?

2007 QEDMO 5th, 5

Tags: algebra , integration , domain , calculus , function , induction , polynomial

Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

2021 Simon Marais Mathematical Competition, B3

Tags: function , calculus , integration

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties. (i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$. (ii) For every real number $x$ and every integer $n \ge 1$ we have \[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]

2022 CMIMC Integration Bee, 13

Tags: calculus , integration

\[\int_{-\infty}^\infty e^{-x^2-4/x^2}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2010 Today's Calculation Of Integral, 575

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

For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.