Found problems: 2215
2010 Today's Calculation Of Integral, 666
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
1980 IMO, 8
Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 2
Let $f(x,\ y)=\frac{x+y}{(x^2+1)(y^2+1)}.$
(1) Find the maximum value of $f(x,\ y)$ for $0\leq x\leq 1,\ 0\leq y\leq 1.$
(2) Find the maximum value of $f(x,\ y),\ \forall{x,\ y}\in{\mathbb{R}}.$
2004 USA Team Selection Test, 3
Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
1999 China Team Selection Test, 2
For a fixed natural number $m \geq 2$, prove that
[b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\]
[b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
2011 Today's Calculation Of Integral, 724
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
1970 IMO Longlists, 47
Given a polynomial
\[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\]
where $a, b, c \neq 0$, prove that $P(x)$ is divisible by
\[Q(x) = abx^2 + (a^2 + b^2)x + ab\]
and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.
2024 UMD Math Competition Part II, #4
Prove for every positive integer $n{:}$
\[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]
1959 Putnam, A7
If $f$ is a real-valued function of one real variable which has a continuous derivative on the closed interval $[a,b]$ and for which there is no $x\in [a,b]$ such that $f(x)=f'(x)=0$, then prove that there is a function $g$ with continuous first derivative on $[a,b]$ such that $fg'-f'g$ is positive on $[a,b].$
2024 CMIMC Integration Bee, 3
\[\int_0^1 \frac{\log(x)}{\sqrt x}\mathrm dx\]
[i]Proposed by Robert Trosten[/i]
2009 Serbia Team Selection Test, 1
Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.
2006 Flanders Math Olympiad, 4
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that
\[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]
2011 Today's Calculation Of Integral, 693
Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$
created by kunny
2009 Today's Calculation Of Integral, 439
Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$.
Note that you may not directively use double integral here for Japanese high school students who don't study it.
2006 Petru Moroșan-Trident, 2
Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $
[i]Dan Negulescu[/i]
2011 Spain Mathematical Olympiad, 2
Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.
2011 Tokyo Instutute Of Technology Entrance Examination, 1
Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$.
For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$
2011 Today's Calculation Of Integral, 718
Find $\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).$
2011 Today's Calculation Of Integral, 748
Evaluate the following integrals.
(1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$
(2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$
2008 AIME Problems, 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2012 Today's Calculation Of Integral, 841
Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$
2011 Iran Team Selection Test, 10
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds.
2007 All-Russian Olympiad, 6
Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots?
[i]N. Agakhanov, I. Bogdanov[/i]
2006 Moldova National Olympiad, 12.2
Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$
2003 CentroAmerican, 3
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.