Olympiad problems

Today's calculation of integrals, 769

Tags: algebra , integration , calculus , domain , ellipse , conic , function

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

2009 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , logarithm , function , integration

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

2005 Today's Calculation Of Integral, 73

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

Find the minimum value of $\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx$

2010 Balkan MO Shortlist, N2

Tags: calculus , number theory , integration

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

2012 Today's Calculation Of Integral, 814

Tags: calculus , calculus computations , integration , geometry

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

An engineer is designing an engine. Each cycle, it ignites a negligible amount of fuel, releasing $ 2000 \, \text{J} $ of energy into the cubic decimeter of air, which we assume here is gaseous nitrogen at $ 20^\circ \, \text{C} $ at $ 1 \, \text{atm} $ in the engine in a process which we can regard as instantaneous and isochoric. It then expands adiabatically until its pressure once again reaches $ 1 \, \text{atm} $, and shrinks isobarically until it reaches its initial state. What is the efficiency of this engine? [i]Problem proposed by B. Dejean[/i]

2012 Today's Calculation Of Integral, 854

Tags: analytic geometry , ellipse , conic , geometry , integration , limit , calculus

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 $\lim_{n\to\infty} S_n$.

2011 Today's Calculation Of Integral, 768

Tags: symmetry , calculus , 3d geometry , limit , sphere , geometry , integration

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2011 Today's Calculation Of Integral, 726

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

Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

2019 Jozsef Wildt International Math Competition, W. 3

Tags: calculus , trigonometry , integration

Compute $$\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx$$

1985 ITAMO, 10

Tags: integration , calculus , floor function , function

How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

2008 All-Russian Olympiad, 5

Tags: inequalities , calculus , integration , number theory

The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?

2023 CMIMC Integration Bee, 13

Tags: calculus , integration

\[\int_0^1 2^{\sqrt x}\log^2(2)+\log^2(1+x)\,\mathrm dx\] [i]Proposed by Thomas Lam[/i]

1991 Arnold's Trivium, 74

Tags: trigonometry , integration

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

2008 Moldova MO 11-12, 8

Tags: trigonometry , real analysis , logarithm , integration

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2007 Today's Calculation Of Integral, 237

Tags: logarithm , absolute value , calculus computations , integration , calculus

Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

2005 Putnam, B3

Tags: algebra , logarithm , absolute value , functional equation , function , integration

Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that \[ f'\left(\frac ax\right)=\frac x{f(x)} \] for all $x>0.$

2022 CMIMC Integration Bee, 9

Tags: integration , calculus

\[\int_e^{e^2} (\log(x))^{\log(x)}(2+\log(\log(x)))\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2011 Putnam, B5

Tags: calculus , function , integration , inequalities , vector

Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$ \[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\] Prove there is a constant $B>0$ such that for all $n,$ \[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]

2005 Today's Calculation Of Integral, 3

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

Calculate the following indefinite integrals. [1] $\int \sin x\sin 2x dx$ [2] $\int \frac{e^{2x}}{e^x-1}dx$ [3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$ [4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$ [5] $\int \frac{e^x}{e^x+1}dx$

1956 AMC 12/AHSME, 25

Tags: calculus , integration

The sum of all numbers of the form $ 2k \plus{} 1$, where $ k$ takes on integral values from $ 1$ to $ n$ is: $ \textbf{(A)}\ n^2 \qquad\textbf{(B)}\ n(n \plus{} 1) \qquad\textbf{(C)}\ n(n \plus{} 2) \qquad\textbf{(D)}\ (n \plus{} 1)^2 \qquad\textbf{(E)}\ (n \plus{} 1)(n \plus{} 2)$

2008 AIME Problems, 3

Tags: calculus , modular arithmetic , integration

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

2012 Today's Calculation Of Integral, 794

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

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

2021 JHMT HS, 9

Tags: function , calculus , integration

Define a sequence $\{ a_n \}_{n=0}^{\infty}$ by $a_0 = 1,$ $a_1 = 8,$ and $a_n = 2a_{n-1} + a_{n-2}$ for $n \geq 2.$ The infinite sum \[ \sum_{n=1}^{\infty} \int_{0}^{2021\pi/14} \sin(a_{n-1}x)\sin(a_nx)\,dx \] can be expressed as a common fraction $\tfrac{p}{q}.$ Compute $p + q.$

2008 Alexandru Myller, 3

Tags: integration , real analysis , calculus , integral , function , inequalities

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]