Olympiad problems

2013 Hitotsubashi University Entrance Examination, 3

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

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 $S$.

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

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]

1999 Finnish National High School Mathematics Competition, 1

Tags: calculus , integration , modular arithmetic , number theory

Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$

1985 IMO Shortlist, 6

Tags: algebra , sequence , inequality , calculus , recurrence relation , inequalities

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

2013 Today's Calculation Of Integral, 895

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

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

1958 February Putnam, A5

Tags: calculus , integration , equation

Show that the integral equation $$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$ has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$

1990 APMO, 3

Tags: calculus , trigonometry , geometry

Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$. For which of these triangles is the product of its altitudes a maximum?

2007 Today's Calculation Of Integral, 179

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2012 Today's Calculation Of Integral, 805

Tags: calculus , integration , inequalities , trigonometry , calculus computations

Prove the following inequalities: (1) For $0\leq x\leq 1$, \[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\] (2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$

2005 Today's Calculation Of Integral, 46

Tags: calculus , integration , calculus computations , logarithm

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

2007 All-Russian Olympiad, 6

Tags: polynomial , calculus , integration , quadratic , algebra

Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots? [i]N. Agakhanov, I. Bogdanov[/i]

2021 JHMT HS, 2

Tags: calculus , integration , algebra , floor function , function

Compute the smallest positive integer $n$ such that $\int_{0}^{n} \lfloor x\rfloor\,dx$ is at least $2021.$

2010 N.N. Mihăileanu Individual, 2

Tags: function , calculus , integration , real analysis

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality $$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$ for any real number $ x. $ Prove that $ f=0. $ [i]Nelu Chichirim[/i]

2004 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3d geometry , sphere , calculus , inequalities

A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.

1986 AMC 12/AHSME, 29

Tags: inequalities , calculus , integration , triangle inequality

Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $

2010 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , factorial , integration

Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.

2012 Canadian Mathematical Olympiad Qualification Repechage, 6

Tags: calculus , derivative , quadratic , algebra

Determine whether there exist two real numbers $a$ and $b$ such that both $(x-a)^3+ (x-b)^2+x$ and $(x-b)^3 + (x-a)^2 +x$ contain only real roots.

2009 Today's Calculation Of Integral, 511

Tags: calculus , integration , derivative , trigonometry , calculus computations

Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

2003 Romania National Olympiad, 3

Tags: function , calculus , integration

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that $$ xf(x)\ge \int_0^x f(t)dt , $$ for all real numbers $ x. $ Prove that [b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $ [b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant. [i]Mihai Piticari[/i]