Olympiad problems

2014 BMT Spring, 19

Tags: integration , calculus

Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.

1999 Putnam, 2

Tags: polynomial , quadratic , derivative , algebra , calculus

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

2005 Putnam, B6

Tags: matrix , linear algebra , integration , algebra , polynomial , calculus

Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that \[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]

2002 Vietnam Team Selection Test, 2

Tags: diophantine equation , parameterization , derivative , calculus , polynomial , algebra

Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

2011 ISI B.Stat Entrance Exam, 4

Tags: calculus computations , calculus , function , integration , inequalities

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

2005 AMC 10, 14

Tags: calculus , integration

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

1991 Arnold's Trivium, 17

Tags: 3d geometry , sphere , geometry , trigonometry , integration , calculus , function

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.

2005 Today's Calculation Of Integral, 6

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

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

2021 JHMT HS, 1

Tags: optimization , calculus , function

The value of $x$ in the interval $[0, 2\pi]$ that minimizes the value of $x + 2\cos x$ can be written in the form $a\pi/b,$ where $a$ and $b$ are relatively prime positive integers. Compute $a + b.$

1960 AMC 12/AHSME, 11

Tags: vieta , calculus , integration

For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

2022 CMIMC Integration Bee, 5

Tags: integration , calculus

\[\int \frac{1}{(1+x)\sqrt{x}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2011 Tokyo Instutute Of Technology Entrance Examination, 1

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

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

2013 District Olympiad, 1

Tags: limit , calculus computations , integration , calculus

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

1969 IMO Shortlist, 56

Tags: decimal representation , inequality , calculus

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

2011 Today's Calculation Of Integral, 678

Tags: trigonometry , calculus , calculus computations , integration

Evaluate \[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\] [i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]

2004 German National Olympiad, 6

Tags: calculus , integration , combinatorics

Is there a circle which passes through five points with integer co-ordinates?

2021 The Chinese Mathematics Competition, Problem 1

Tags: calculus

Evaluate $\lim_{x \to +\infty}\sqrt{x^2+x+1}\frac{x-ln(e^x+x)}{x}$.

1991 Arnold's Trivium, 51

Tags: trigonometry , integration , calculus

Calculate the integral \[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]

2009 Today's Calculation Of Integral, 497

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

Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

2005 District Olympiad, 4

Tags: group theory , algebra , domain , integration , calculus , abstract algebra , function

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

2013 Today's Calculation Of Integral, 884

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

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

2024 CMIMC Integration Bee, 10

Tags: integration , calculus

\[\int_{-1}^1 \sqrt[3]{x}\log(1+e^x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Today's calculation of integrals, 851

Tags: geometric series , limit , calculus , integration , calculus computations , trigonometry , function

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

1984 AMC 12/AHSME, 29

Tags: quadratic , analytic geometry , slope , algebra , graphing lines , calculus , trigonometry

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

2011 Today's Calculation Of Integral, 700

Tags: calculus computations , trigonometry , calculus , integration

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]