Olympiad problems

1988 Flanders Math Olympiad, 4

Tags: calculus , derivative , geometry , trigonometry , function , inequalities , quadratic

Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.

2009 Albania Team Selection Test, 2

Tags: function , algebra , polynomial , induction , calculus , derivative , limit

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

2021 JHMT HS, 7

Tags: calculus

In three-dimensional space, let $\mathcal{S}$ be the surface consisting of all points $(x, y, 0)$ satisfying $x^2 + 1 \leq y \leq 2,$ and let $A$ be the point $(0, 0, 900).$ Compute the volume of the solid obtained by taking the union of all line segments with endpoints in $\mathcal{S} \cup \{A\}.$

2011 Today's Calculation Of Integral, 722

Tags: calculus , integration , function , calculus computations

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

1979 Spain Mathematical Olympiad, 8

Tags: real analysis , derivative , calculus

Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$ Express the numerical value of its derivative of order $325$ for $x = 0$.

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 \[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]

1972 Poland - Second Round, 6

Tags: algebra , calculus

Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

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

2021 The Chinese Mathematics Competition, Problem 10

Tags: calculus

Let ${a_n}$ and ${b_n}$ be positive real sequence that satisfy the following condition: (i) $a_1=b_1=1$ (ii) $b_n=a_n b_{n-1}-2$ (iii) $n$ is an integer larger than $1$. Let ${b_n}$ be a bounded sequence. Prove that $\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}$ converges. Find the value of the sum.

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 $[-1,\ 1]$.

2009 Greece National Olympiad, 3

Tags: calculus , inequalities

Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?

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)$.