Olympiad problems

2009 Today's Calculation Of Integral, 442

Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

2005 Alexandru Myller, 2

Tags: real analysis , calculus , integration , function

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

2007 Today's Calculation Of Integral, 173

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

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

1980 Vietnam National Olympiad, 2

Tags: algebra , polynomial , integration , calculus

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

1975 Miklós Schweitzer, 5

Tags: limit , integration , function , real analysis

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

2010 Today's Calculation Of Integral, 647

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

Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

Today's calculation of integrals, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2017 VJIMC, 4

Tags: integration , real analysis , calculus

Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that \[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]

1973 Miklós Schweitzer, 6

Tags: integration , function , real analysis , search

If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.\] [i]Z. Daroczy[/i]

2013 Today's Calculation Of Integral, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2006 Putnam, B5

Tags: function , algebra , integration , quadratic , calculus , inequalities

For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

2009 Today's Calculation Of Integral, 484

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

Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

2021 CMIMC Integration Bee, 12

Tags: integration , calculus

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

2010 Today's Calculation Of Integral, 632

Tags: calculus computations , integration , floor function , trigonometry , calculus , limit , ceiling function

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

2006 IberoAmerican Olympiad For University Students, 7

Tags: function , algebra , modular arithmetic , search , trigonometry , calculus , integration

Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$. Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.

1992 India National Olympiad, 6

Tags: polynomial , integration , calculus , algebra

Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.

1960 AMC 12/AHSME, 28

Tags: calculus , integration

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$ $\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $

2009 Today's Calculation Of Integral, 460

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

$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.

2011 Albania Team Selection Test, 2

Tags: geometry , perimeter , trigonometry , integration , algorithm , inradius , calculus

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

PEN G Problems, 16

Tags: inequalities , algebra , integration , irrational number , polynomial

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

2006 District Olympiad, 1

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

Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]

2013 Today's Calculation Of Integral, 887

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

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2011 Mediterranean Mathematics Olympiad, 3

Tags: trigonometry , tetrahedron , 3d geometry , sphere , integration , geometry

A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.) Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.

2007 Estonia Math Open Junior Contests, 4

Tags: perimeter , ratio , integration , calculus , number theory , geometry

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

2012 Today's Calculation Of Integral, 783

Tags: calculus computations , integration , calculus

Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited