Olympiad problems

2000 Romania Team Selection Test, 3

Tags: geometry , 3d geometry , sphere , function , calculus , topology , analytic geometry

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

2005 Today's Calculation Of Integral, 79

Tags: calculus , integration , geometry , algebra , function , domain , factorial

Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

2011 Bogdan Stan, 4

Tags: function , calculus , derivative , integrability , convexity , side-continuity

Let be an open interval $ I $ and a convex function $ f:I\longrightarrow\mathbb{R} . $ Prove that the lateral derivatives of $ f $ are left-continuous on $ \mathbb{R} $ and also right-continuous on $ \mathbb{R} . $ [i]Marin Tolosi[/i]

2005 Indonesia MO, 1

Tags: calculus , integration , inequalities , combinatorics

Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

2000 Moldova National Olympiad, Problem 4

Tags: integration , function , calculus

Let $f:[0,1]\to\mathbb R$ be a continuous function such that $\int^1_0x^mf(x)dx=0$ for $m=0,1,\ldots,1999$. Prove that $f$ has at least $2000$ zeroes on the segment $[0,1]$.

2007 ISI B.Stat Entrance Exam, 3

Tags: calculus , integration , function , calculus computations

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$, \[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]

2008 Moldova Team Selection Test, 4

Tags: polynomial , factorial , calculus , integration , algebra

A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

2011 Today's Calculation Of Integral, 751

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

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

2008 Vietnam National Olympiad, 1

Tags: calculus , limit , algebra , logarithm

Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$

2005 Today's Calculation Of Integral, 11

Tags: calculus , integration , function , domain , complex number , logarithm , algebra

Calculate the following indefinite integrals. [1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$ [2] $\int \frac{e^x}{e^x+e^{a-x}}dx$ [3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$ [4] $\int x\ln (x^2-1)dx$ [5] $\int \frac{2(x+2)}{x^2+4x+1}dx$

2009 Today's Calculation Of Integral, 450

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

Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$

2013 Today's Calculation Of Integral, 889

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

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

2018 Brazil Undergrad MO, 10

Tags: college contest , calculus , limit

How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?

2004 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , geometry

Find the area of the region in the $xy$-plane satisfying $x^6-x^2+y^2 \le 0$.

1969 IMO Shortlist, 56

Tags: decimal representation , inequality , calculus

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

2006 Vietnam National Olympiad, 4

Tags: function , calculus , derivative , limit , algebra

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

2010 Today's Calculation Of Integral, 662

Tags: calculus , integration , geometry , geometric transformation , rotation , inequalities , perimeter

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

2012 Today's Calculation Of Integral, 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 $\lim_{n\to\infty} S_n$.

2007 Today's Calculation Of Integral, 248

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

Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

2010 Today's Calculation Of Integral, 642

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.\] Own

1999 Mexico National Olympiad, 6

Tags: calculus , integration , combinatorics

A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.

2022 JHMT HS, 10

Tags: calculus , trigonometry

The maximum value of \[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \] over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.

2022 CMIMC Integration Bee, 10

Tags: calculus , integration

\[\int_0^1 \frac{(x+1)\log(x)}{x^3-1}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2009 Romania National Olympiad, 1

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

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

1965 AMC 12/AHSME, 9

Tags: parabola , calculus , conic

The vertex of the parabola $ y \equal{} x^2 \minus{} 8x \plus{} c$ will be a point on the $ x$-axis if the value of $ c$ is: $ \textbf{(A)}\ \minus{} 16 \qquad \textbf{(B)}\ \minus{} 4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$