Found problems: 2215
2004 France Team Selection Test, 1
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2005 Today's Calculation Of Integral, 10
Calculate the following indefinite integrals.
[1] $\int (2x+1)\sqrt{x+2}\ dx$
[2] $\int \frac{1+\cos x}{x+\sin x}\ dx$
[3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$
[4] $\int \frac{(x-3)^2}{x^4}\ dx$
[5] $\int \frac{dx}{\tan x}\ dx$
2007 Today's Calculation Of Integral, 174
Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis.
(1) Find the value of $a.$
(2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$
1982 Putnam, B2
Let $A(x,y)$ be the number of points $(m,n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^2+n^2\le x^2+y^2$. Let $g=\sum_{k=1}^\infty e^{-k^2}$. Express
$$\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy$$
as a polynomial in $g$.
2010 Harvard-MIT Mathematics Tournament, 7
Let $a_1$, $a_2$, and $a_3$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of \[\dfrac{|a_1+a_2+a_3|}{\sqrt[3]{|a_1a_2a_3|}}.\]
2012 Today's Calculation Of Integral, 842
Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$
2012 Kyoto University Entry Examination, 1
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points
2010 Tuymaada Olympiad, 2
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.
Show that $\angle BPC > \angle BAC$.
2005 National High School Mathematics League, 3
For positive integer $n$, define $f(n)=\begin{cases}
0, \text{if }n\text{ is a perfect square}\\
\displaystyle \left[\frac{1}{\{\sqrt{n}\}}\right], \text{if }n\text{ is not a perfect square}\\
\end{cases}$.
Find the value of $\sum_{k=1}^{240} f(k)$.
Note: $[x]$ is the integral part of real number $x$, and $\{x\}=x-[x]$.
2010 Today's Calculation Of Integral, 663
Given are the curve $y=x^2+x-2$ and a curve which is obtained by tranfering the curve symmetric with respect to the point $(p,\ 2p)$. Let $p$ change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves.
[i]1978 Nagasaki University entrance exam/Economics[/i]
1973 Poland - Second Round, 3
Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the following conditions:
1. $ f(x+1) = f(x) + 1 $ for each $ x \in \mathbb{R} $,
2. there exists an integer p such that $ f(f(f(O))) = p $. Prove that for every real number $ x $
$$ \lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.$$
where $ x_1 = x $ and $ x_n =f(x_{n-1}) $ for $ n = 2, 3, \ldots $.
2013 BMT Spring, 9
Evaluate the integral
$$\int^1_0\left(\sqrt{(x-1)^3+1}+x^{2/3}-(1-x)^{3/2}-\sqrt[3]{1-x^2}\right)dx$$
2016 BMT Spring, 8
Evaluate the following limit
$$\lim_{x\to 0} (1 + 2x + 3x^2 + 4x^3 +...)^{1/x}$$
1969 Canada National Olympiad, 2
Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.
2003 IMO Shortlist, 3
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
1979 Spain Mathematical Olympiad, 8
Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$
Express the numerical value of its derivative of order $325$ for $x = 0$.
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.
2004 Vietnam Team Selection Test, 2
Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.
2010 ISI B.Math Entrance Exam, 8
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that
$\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.
2010 Today's Calculation Of Integral, 632
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]
2007 Today's Calculation Of Integral, 246
An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$
2011 Romania National Olympiad, 2
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
2014 NIMO Problems, 8
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
[i]Proposed by Evan Chen[/i]
2014 All-Russian Olympiad, 3
If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?
2011 Today's Calculation Of Integral, 764
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$