Olympiad problems

1997 Traian Lălescu, 4

Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].

1968 Putnam, B4

Tags: integration , measure theory

Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $L=\int_{-\infty}^{\infty} f(x) dx$ exists. Show that $$\int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx=L.$$

Today's calculation of integrals, 898

Tags: calculus , integration , logarithm , calculus computations

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2007 Princeton University Math Competition, 3

Tags: algebra , calculus , geometry , integration , analytic geometry , quadratic , quadratic formula

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2021 CMIMC Integration Bee, 9

Tags: calculus , integration

$$\int_1^2\frac{12x^3+12x+12}{2x^4+3x^2+4x}\,dx$$ [i]Proposed by Connor Gordon[/i]

2013 Stanford Mathematics Tournament, 9

Tags: trigonometry , calculus , integration

Evaluate $\int_{0}^{\pi/2}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^4}$.

2010 Today's Calculation Of Integral, 525

Tags: geometry , trigonometry , inequalities , integration , quadratic , calculus , vector

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

2012 Today's Calculation Of Integral, 788

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

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

2012 NIMO Problems, 2

Tags: function , integration , derivative , calculus

For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized? [i]Proposed by Eugene Chen[/i]

1962 AMC 12/AHSME, 22

Tags: calculus , integration

The number $ 121_b$, written in the integral base $ b$, is the square of an integer, for $ \textbf{(A)}\ b \equal{} 10,\text{ only} \qquad \textbf{(B)}\ b \equal{} 10 \text{ and } b \equal{} 5, \text{ only} \qquad \textbf{(C)}\ 2 \leq b \leq 10 \qquad \textbf{(D)}\ b > 2 \qquad \textbf{(E)}\ \text{no value of }b$

2005 Indonesia MO, 1

Tags: integration , calculus , inequalities , combinatorics

Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

1952 Miklós Schweitzer, 9

Tags: real analysis , limit , function , integration

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)

2013 Romanian Master of Mathematics, 4

Tags: reflection , homothety , function , geometry , geometric transformation , ratio , integration

Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?

2010 Today's Calculation Of Integral, 552

Tags: graphing lines , analytic geometry , integration , parabola , calculus , conic , derivative

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

2013 Romanian Masters In Mathematics, 1

Tags: geometry , function , integration , homothety , ratio , geometric transformation , reflection

Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?

PEN H Problems, 46

Tags: calculus , integration , diophantine equation

Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.

2001 VJIMC, Problem 2

Tags: integration , calculus

Let $f:[0,1]\to\mathbb R$ be a continuous function. Define a sequence of functions $f_n:[0,1]\to\mathbb R$ in the following way: $$f_0(x)=f(x),\qquad f_{n+1}(x)=\int^x_0f_n(t)\text dt,\qquad n=0,1,2,\ldots.$$Prove that if $f_n(1)=0$ for all $n$, then $f(x)\equiv0$.

Today's calculation of integrals, 885

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

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2014 Contests, 4

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

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.

2010 Today's Calculation Of Integral, 590

Tags: trigonometry , integration , calculus computations , calculus

Evaluate $ \int_0^{\frac{\pi}{8}} \frac{(\cos \theta \plus{}\sin \theta)^{\frac{3}{2}}\minus{}(\cos \theta \minus{}\sin \theta)^{\frac{3}{2}}}{\sqrt{\cos 2\theta}}\ d\theta$.

2004 District Olympiad, 4

Tags: integration , real analysis , integral , calculus

Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that $$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$ [b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $ [b]b)[/b] Show that if $ a+b=1, $ then $ f $ is constant.

2013 Romania National Olympiad, 4

Tags: integration , logarithm , function , number theory , induction , calculus , inequalities

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

2014 Contests, 2

Tags: inequalities , integration , function , combinatorics

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

2013 Stanford Mathematics Tournament, 8

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

The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

2013 Today's Calculation Of Integral, 869

Tags: integration , trigonometry , calculus computations , calculus

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$