Olympiad problems

2024 CMIMC Integration Bee, 15

Tags: integration , calculus

\[\int_0^\infty 1+\cos\left(\tfrac 1{\sqrt x}\right)-2\cos\left(\tfrac 1{\sqrt {2x}}\right)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Today's Calculation Of Integral, 515

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

Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$. Note that you are not allowed to solve in using partial differentiation here.

2017 BMT Spring, 3

Tags: calculus , integration , algebra

Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$

2010 Today's Calculation Of Integral, 574

Tags: logarithm , calculus , integration , calculus computations

Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2014 India National Olympiad, 4

Tags: calculus , quadratic , analytic geometry , algebra , probability , polynomial , integration

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

1991 Arnold's Trivium, 76

Tags: integration , trigonometry , probability , function

Investigate the behaviour at $t\to\infty$ of the solution of the problem \[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]

Today's calculation of integrals, 880

Tags: geometry , calculus , limit , rotation , geometric transformation , integration , trigonometry

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2005 Today's Calculation Of Integral, 51

Tags: calculus , calculus computations , function , integration

A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

2010 Today's Calculation Of Integral, 578

Tags: calculus , inequalities , logarithm , domain , algebra , function , integration

Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$. \[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\] If necessary, you may use $ \ln 3 \equal{} 1.10$.

2004 VTRMC, Problem 5

Tags: calculus , integration

Let $f(x)=\int^x_0\sin(t^2-t+x)dt$. Compute $f''(x)+f(x)$ and deduce that $f^{(12)}(0)+f^{(10)}(0)=0$.

1998 IMC, 3

Tags: limit , real analysis , integration

Let $f(x)=2x(1-x), x\in\mathbb{R}$ and denote $f_n=f\circ f\circ ... \circ f$, $n$ times. (a) Find $\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx$. (b) Now compute $\int^1_0 f_n(x)dx$.

2010 Today's Calculation Of Integral, 557

Tags: integration , limit , calculus , floor function , trigonometry , function , analytic geometry

Find the folllowing limit. \[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]

2014 Online Math Open Problems, 29

Tags: 3d geometry , tetrahedron , geometry , calculus , circumcircle , integration

Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$. [i]Proposed by Sammy Luo and Evan Chen[/i]

2007 Gheorghe Vranceanu, 4

Tags: real analysis , cesaro-stolz , sequence , integral limit , function , integration , calculus

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$

2011 Today's Calculation Of Integral, 727

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

For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

2011 Today's Calculation Of Integral, 702

Tags: calculus computations , calculus , integration , function

$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$

1989 Putnam, B6

Tags: expected value , probability , integration , calculus

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

2010 ELMO Shortlist, 4

Tags: combinatorics , integration

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

2005 Today's Calculation Of Integral, 68

Tags: calculus , logarithm , calculus computations , derivative , integration

Find the minimum value of $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

1989 IMO Shortlist, 8

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

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 NIMO Problems, 3

Tags: calculus , integration

In land of Nyemo, the unit of currency is called a [i]quack[/i]. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay oﬀ $125$ quacks using these coins? [i]Proposed by Aaron Lin[/i]

1972 Canada National Olympiad, 10

Tags: calculus , geometric sequence , integration , ratio , algebra

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2010 Today's Calculation Of Integral, 553

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

Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.

2021 JHMT HS, 2

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

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

2009 Today's Calculation Of Integral, 505

Tags: calculus computations , trigonometry , integration , calculus

In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.