Olympiad problems

2009 Today's Calculation Of Integral, 405

Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.

1996 South africa National Olympiad, 3

Tags: calculus , integration , trigonometry , geometry

The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

2009 Today's Calculation Of Integral, 429

Tags: calculus , integration , trigonometry , calculus computations

Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.

2009 Today's Calculation Of Integral, 498

Tags: calculus , integration , function , calculus computations

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

2014 Miklós Schweitzer, 2

Tags: inequalities , function , integration , 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 $I_i\cap I_j\neq \emptyset$.

2005 Today's Calculation Of Integral, 39

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

Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$. \[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]

PEN P Problems, 12

Tags: function , floor function , inequalities , calculus , integration , logarithm

The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

2013 District Olympiad, 3

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

Problem 3. Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that: (a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

2006 APMO, 2

Tags: calculus , integration , ratio , induction , algorithm , algebra

Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.

2005 IMC, 3

Tags: integration , inequalities

3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$

2013 Saint Petersburg Mathematical Olympiad, 1

Tags: trigonometry , floor function , calculus , integration , algebra

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov

2022 CMIMC Integration Bee, 3

Tags: calculus , integration

\[\int_0^1 x\sqrt[4]{1-x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2005 Morocco TST, 3

Tags: quadratic , calculus , integration , number theory

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

2001 Romania National Olympiad, 4

Tags: function , limit , integration , logarithm , real analysis

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

1970 IMO Longlists, 25

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

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

Today's calculation of integrals, 864

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

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2012 Today's Calculation Of Integral, 786

Tags: calculus , integration , limit , induction , algebra , polynomial , probability

For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

2022 SEEMOUS, 2

Tags: functional equation , continuity , palic , differentiation , integration , analysis

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

2009 Today's Calculation Of Integral, 437

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

2005 Today's Calculation Of Integral, 31

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

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

2022 CMIMC Integration Bee, 5

Tags: calculus , integration

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

2010 Today's Calculation Of Integral, 664

Tags: calculus , integration , trigonometry , calculus computations

For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$. Find $I_1+I_2+I_3+I_4$. [i]1992 University of Fukui entrance exam/Medicine[/i]

1964 Putnam, A2

Tags: integration , system of equations

Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that $$\int_{0}^{1} f(x)\; dx =1, $$ $$\int_{0}^{1} xf(x)\; dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$ where $\alpha$ is a given real number.

1976 Miklós Schweitzer, 8

Tags: algebra , polynomial , integration , real analysis

Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$. [i]J. Szabados[/i]

Today's calculation of integrals, 895

Tags: calculus , integration , analytic geometry , parabola , calculus computations , conic , geometry

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.