Found problems: 2215
2018 CMI B.Sc. Entrance Exam, 1
Answer the following questions :
$\textbf{(a)}~$ A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2,\cdots, a_k$, each $a_i>1$, such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$$ Show that if $k$ is stable, then $(k+1)$ is also stable. Using this or otherwise, find all stable numbers.
$\textbf{(b)}$ Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}$$ whenever the above maximum is finite.
For the function $f(x)=\ln x$, determine the set of points for which $f^*$ is defined and find an expression for $f^*(y)$ involving only $y$ and constants.
2013 Today's Calculation Of Integral, 877
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$
Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$
2013 Singapore MO Open, 5
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.
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}$.
2011 Today's Calculation Of Integral, 766
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 $\int_0^{\pi} \{f(x)\}^3dx.$
2006 Harvard-MIT Mathematics Tournament, 1
A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?
2020 CHMMC Winter (2020-21), 6
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[i]Note: for a complex number $z = re^{i \theta}$ for reals $r > 0, 0 \le \theta < 2\pi$, we define $z^{n} = r^{n} e^{i \theta n}$ for all positive reals $n$.[/i]
2013 AMC 10, 18
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
2010 Harvard-MIT Mathematics Tournament, 5
Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.
2009 Today's Calculation Of Integral, 426
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$.
2007 IMO Shortlist, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2006 Victor Vâlcovici, 2
Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that
[b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $
[b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $
2012 AIME Problems, 8
Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((0,10)--(4,13)--(14,13)--(10,10));
draw((10,0)--(14,3)--(14,13));
draw((0,0)--(4,3)--(4,13), dashed);
draw((4,3)--(14,3), dashed);
dot((0,0));
dot((0,10));
dot((10,10));
dot((10,0));
dot((4,3));
dot((14,3));
dot((14,13));
dot((4,13));
dot((14,8));
dot((5,0));
label("A", (0,0), SW);
label("B", (10,0), S);
label("C", (14,3), E);
label("D", (4,3), NW);
label("E", (0,10), W);
label("F", (10,10), SE);
label("G", (14,13), E);
label("H", (4,13), NW);
label("M", (5,0), S);
label("N", (14,8), E);
[/asy]
2010 Albania National Olympiad, 3
[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.
[b](b)[/b]What is the smallest area possible of pentagons with integral coordinates.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.
2011 Harvard-MIT Mathematics Tournament, 1
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.
2024 ISI Entrance UGB, P4
Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows:
$$g(x) = \begin{cases}
f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\
0 &\text{if} ~ x = 0.
\end{cases}$$
Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]
2011 Today's Calculation Of Integral, 683
Evaluate $\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx$.
[i]2011 Kyoto University entrance exam/Science, Problem 1B[/i]
1979 Spain Mathematical Olympiad, 5
Calculate the definite integral
$$\int_2^4 \sin ((x-3)^3) dx$$
2013 Hitotsubashi University Entrance Examination, 3
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$.
(1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$.
(2) If $q=p+1$, then find the minimum value of $S$.
(3) If $pq=-1$, then find the minimum value of $S$.
2018 Ramnicean Hope, 1
Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation
$$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$
Calculate $ \int_{-2019}^{2019}f(x)dx . $
[i]Constantin Rusu[/i]
2009 Today's Calculation Of Integral, 486
Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$
Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions.
(1) Express the coordiante of $ H$ in terms of $ t$.
(2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$.
(3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.
2013 Today's Calculation Of Integral, 897
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
2007 Moldova National Olympiad, 11.8
The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.
2009 Today's Calculation Of Integral, 473
For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\
r^2 & (r\leq x\leq l \plus{} r)\quad \\
(x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$
Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.
1999 Harvard-MIT Mathematics Tournament, 6
Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.