Found problems: 1687
2007 Princeton University Math Competition, 8
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?
2011 Canadian Open Math Challenge, 8
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
Today's calculation of integrals, 885
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$
2011 Vietnam Team Selection Test, 1
A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$
$(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$
$(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$
2009 Today's Calculation Of Integral, 515
Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$.
Note that you are not allowed to solve in using partial differentiation here.
2023 CMIMC Integration Bee, 14
\[\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
2015 Kyoto University Entry Examination, 5
5. Let $a,b,c,d,e$ be positive rational numbers. Consider integral expressions
$f(x)=ax^2+bx+c$
$g(x)=dx+e$
Put $\frac{f(n)}{g(n)}$ an integer for all positive integers $n$. Then, show that $f(x)$ is dividible by $g(x)$.
2005 AMC 12/AHSME, 11
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
$ \textbf{(A)}\ 41\qquad
\textbf{(B)}\ 42\qquad
\textbf{(C)}\ 43\qquad
\textbf{(D)}\ 44\qquad
\textbf{(E)}\ 45$
2019 Jozsef Wildt International Math Competition, W. 7
If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$
2011 Today's Calculation Of Integral, 732
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
2007 Today's Calculation Of Integral, 206
Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$
2005 Today's Calculation Of Integral, 47
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
1978 Putnam, A3
Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.
2009 Today's Calculation Of Integral, 481
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$.
Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$
2010 Today's Calculation Of Integral, 661
Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$.
(1) Find the 36th term.
(2) Find $\int x^2\ln x\ dx$.
(3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part?
If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$.
[i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]
2017 BMT Spring, 8
The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$.
(Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)
2010 Today's Calculation Of Integral, 546
Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.
2007 Today's Calculation Of Integral, 235
Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.
2013 Romania National Olympiad, 3
Given $a\in (0,1)$ and $C$ the set of increasing functions
$f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine:
$(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$
$(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$
1961 AMC 12/AHSME, 4
Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 . . .$. If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under:
${{ \textbf{(A)}\ \text{Addition}\qquad\textbf{(B)}\ \text{Multiplication} \qquad\textbf{(C)}\ \text{Division} \qquad\textbf{(D)}\ \text{Extraction of a positive integral root} }\qquad\textbf{(E)} \text{None of these} } $
2021 CMIMC Integration Bee, 8
$$\int\left(\frac{x-1}{x^2+1}\right)^2e^x\,dx$$
[i]Proposed by Connor Gordon[/i]
2011 Today's Calculation Of Integral, 693
Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$
created by kunny
2005 Today's Calculation Of Integral, 88
A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$
Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$
1949 Miklós Schweitzer, 7
Find the complex numbers $ z$ for which the series
\[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\]
converges and find its sum.
2012 Today's Calculation Of Integral, 834
Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.