Olympiad problems

2002 India National Olympiad, 3

Tags: algebra , calculus , polynomial , inequalities , inequality

If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.

1989 IMO Longlists, 20

Tags: rectangle , number theory , integration , geometry , calculus

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

2014 Contests, 903

Tags: real analysis , calculus , integration , function , geometric series , calculus computations

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

1982 IMO Longlists, 19

Tags: inequality , geometric progression , calculus

Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

2012 Today's Calculation Of Integral, 856

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

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2010 Today's Calculation Of Integral, 606

Tags: geometry , calculus , integration , calculus computations

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

2013 District Olympiad, 4

Tags: calculus computations , function , calculus , limit

Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function. a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$. b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.

1996 China National Olympiad, 2

Tags: inequalities , trigonometry , derivative , integration , calculus

Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

1958 AMC 12/AHSME, 46

Tags: function , algebra , calculus , derivative , domain

For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression \[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} \] has: $ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{\minus{}1}$

2006 ISI B.Math Entrance Exam, 5

Tags: calculus , rectangle , integration , geometry

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

2024-25 IOQM India, 10

Tags: calculus , integration

Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a,b,$ and $c$ which satisfy $$a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$$

2024 Romania National Olympiad, 1

Tags: integration , function , calculus , real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)+\sin(f(x)) \ge x,$ for all $x \in \mathbb{R}.$ Prove that $$\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.$$

2007 Today's Calculation Of Integral, 212

Tags: integration , trigonometry , calculus , calculus computations

For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.

2010 Today's Calculation Of Integral, 545

Tags: function , calculus , calculus computations , integration

(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

2001 USA Team Selection Test, 2

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

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

2012 Uzbekistan National Olympiad, 2

Tags: calculus , number theory , modular arithmetic , integration

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: logarithm , integration , calculus computations , calculus

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

PEN R Problems, 12

Tags: integration , calculus , analytic geometry , the geometry of numbers

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

2011 China Team Selection Test, 2

Tags: number theory , combinatorics , function , calculus

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

2012 Today's Calculation Of Integral, 789

Tags: function , calculus computations , calculus , integration

Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$

2021 Alibaba Global Math Competition, 15

Tags: integration , calculus , topology , analysis , manifold , differential geometry

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

2005 Today's Calculation Of Integral, 64

Tags: trigonometry , integration , algebra , calculus , polynomial , calculus computations

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

1986 Traian Lălescu, 2.3

Tags: function , inequalities , calculus , derivative , ftc , rolle theorem , integral

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

2008 Gheorghe Vranceanu, 1

Tags: function , inverse calculus , calculus , primitive

Find the $ \mathcal{C}^1 $ class functions $ f:[0,1]\longrightarrow\mathbb{R} $ satisfying the following three clauses: $ \text{(i) } f(0)=0 $ $ \text{(ii) } \text{Im} f'\subset (0,1] $ $ \text{(iii) }F(1)-\frac{\left( f(1) \right)^3}{3} =F(0)=0, $ where $ F $ is a primitive of $ f. $

1979 IMO Longlists, 29

Tags: calculus , inequality , maximization , optimization , lagrange

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.