Found problems: 1687
2004 District Olympiad, 4
Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that
$$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$
[b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $
[b]b)[/b] Show that if $ a+b=1, $ then $ f $ is constant.
2024 CMIMC Integration Bee, 1
\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2021 CMIMC Integration Bee, 9
$$\int_1^2\frac{12x^3+12x+12}{2x^4+3x^2+4x}\,dx$$
[i]Proposed by Connor Gordon[/i]
2009 Today's Calculation Of Integral, 517
Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.
2013 SEEMOUS, Problem 1
Find all continuous functions $f:[1,8]\to\mathbb R$, such that
$$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$
2009 Today's Calculation Of Integral, 404
Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
2010 Today's Calculation Of Integral, 622
For $0<k<2$, consider two curves $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$
Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$.
Find the minimum value of $S(k).$
[i]2010 Nagoya Institute of Technology entrance exam[/i]
1983 Putnam, B1
Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.
2007 F = Ma, 31
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
Find the ratio $L/d$.
$ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $
2012 USAMO, 6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)
Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?
2014 Contests, 4
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
2013 Bogdan Stan, 2
Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties:
$ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $
$ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $
Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $
[i]Cristinel Mortici[/i]
2012 District Olympiad, 1
Let $a,b,c$ three positive distinct real numbers. Evaluate:
\[\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx\]
Today's calculation of integrals, 856
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.
2010 Today's Calculation Of Integral, 637
For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions:
(1) Calculate $I_{n+2}+I_n.$
(2) Evaluate the values of $I_1,\ I_2$ and $I_3.$
1978 Niigata university entrance exam
2022 CMIMC Integration Bee, 15
\[\int_0^\infty 1+\frac{2}{\sqrt[x]{8}}-\frac{3}{\sqrt[x]{4}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2007 Today's Calculation Of Integral, 244
A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
2012 Today's Calculation Of Integral, 821
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
2007 Princeton University Math Competition, 4
Find the sum of the reciprocals of the positive integral factors of $84$.
2012 Today's Calculation Of Integral, 820
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$
2010 Today's Calculation Of Integral, 565
Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.
2014 Contests, 902
For $a\geq 0$, find the minimum value of $S(a)=\int_0^1 |x^2+2ax+a^2-1|\ dx.$
2011 Poland - Second Round, 3
There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial:
\[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\]
is divisible by $P(x)-Q(x)$.
2023 CMIMC Integration Bee, 9
\[\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx\]
[i]Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko[/i]
2016 District Olympiad, 2
Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that
$$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$
with equality if and only if $ 1 $ is also a period of $ f. $