Olympiad problems

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

1992 Putnam, A4

Tags: derivative , limit , trigonometry , search , function , calculus

Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$

1955 AMC 12/AHSME, 10

Tags: calculus , slope , function , graphing lines , analytic geometry

How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

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]

1967 IMO Shortlist, 3

Tags: trigonometry , taylor series , inequality , trigonometric inequality , calculus

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

2007 Today's Calculation Of Integral, 237

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

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

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] $\int \frac{e^x}{e^x+1}dx$

2014 Contests, 4

Tags: three variable inequality , calculus , inequalities

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2000 Romania Team Selection Test, 3

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

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]

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)$

2021 Simon Marais Mathematical Competition, A1

Tags: conic , parabola , calculus

Let $a, b, c$ be real numbers such that $a \neq 0$. Consider the parabola with equation \[ y = ax^2 + bx + c, \] and the lines defined by the six equations \begin{align*} &y = ax + b, \quad & y = bx + c, \qquad \quad & y = cx + a, \\ &y = bx + a, \quad & y = cx + b, \qquad \quad & y = ax + c. \end{align*} Suppose that the parabola intersects each of these lines in at most one point. Determine the maximum and minimum possible values of $\frac{c}{a}$.

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]

2010 Harvard-MIT Mathematics Tournament, 1

Tags: polynomial , calculus , derivative , algebra

Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.

2010 Today's Calculation Of Integral, 554

Tags: calculus , integration , logarithm , calculus computations

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

2025 Romania National Olympiad, 3

Tags: derivative , function , calculus , real analysis

Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent: a) $f$ is differentiable, with continuous first derivative. b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.

2005 Today's Calculation Of Integral, 29

Tags: calculus , calculus computations , trigonometry , integration

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]