Olympiad problems

2017 Azerbaijan EGMO TST, 4

Tags: calculus , integration , induction , modular arithmetic , number theory

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2009 Today's Calculation Of Integral, 446

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

1999 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , trigonometry , function

Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]

For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[ \int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn \] for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposed by Evan Chen[/i]

2009 Today's Calculation Of Integral, 426

Tags: calculus , integration , algebra , polynomial , calculus computations

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

2001 India Regional Mathematical Olympiad, 3

Tags: floor function , calculus , integration

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

2011 Today's Calculation Of Integral, 722

Tags: calculus , integration , function , calculus computations

Find the continuous function $f(x)$ such that : \[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]

2007 Today's Calculation Of Integral, 229

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

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

2005 Today's Calculation Of Integral, 32

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

2022 VJIMC, 1

Tags: calculus , integration , inequalities , function

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

2010 Today's Calculation Of Integral, 531

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

(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that \[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\] Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$, Prove that there exists $ \theta$ such that \[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]

2020 Jozsef Wildt International Math Competition, W32

Tags: integration , calculus

Compute the quadruple integral $$A=\frac1{\pi^2}\int_{[0,1]^2\times[-\pi,\pi]^2}ab\sqrt{a^2+b^2-2ab\cos(x-y)}dadbdxdy$$ [i]Proposed by Moubinool Omarjee[/i]

2012 Today's Calculation Of Integral, 794

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

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 $0\leq x\leq \pi$.

2005 Today's Calculation Of Integral, 35

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

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

2004 IMC, 5

Tags: integration , inequalities , calculus , logarithm

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

2011 Today's Calculation Of Integral, 710

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

Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2007 Today's Calculation Of Integral, 181

Tags: calculus , integration , trigonometry , function , derivative , analytic geometry , logarithm

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

2012 Today's Calculation Of Integral, 834

Tags: calculus , integration , geometry , calculus computations

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

2010 Today's Calculation Of Integral, 574

Tags: calculus , integration , calculus computations , logarithm

Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2010 Today's Calculation Of Integral, 647

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

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

2004 China Team Selection Test, 3

Tags: algebra , polynomial , calculus , integration , number theory

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2011 Today's Calculation Of Integral, 680

Tags: calculus , integration , calculus computations

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

2007 Princeton University Math Competition, 6

Tags: probability , calculus , integration , geometry , trapezoid , logarithm

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

2010 Today's Calculation Of Integral, 547

Tags: calculus , integration , function , derivative , real analysis , calculus computations , logarithm

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2013 Today's Calculation Of Integral, 863

Tags: calculus , integration , trigonometry , limit , derivative , inequalities , calculus computations

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$