Found problems: 2215
2006 ISI B.Math Entrance Exam, 5
A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???
2009 Today's Calculation Of Integral, 403
Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.
2013 Kosovo National Mathematical Olympiad, 2
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
2009 Today's Calculation Of Integral, 457
Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$
1989 Putnam, A6
Let $\alpha=1+a_1x+a_2x^2+\ldots$ be a formal power series with coefficients in the field of two elements. Let
$$a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}$$(For example, $a_{36}=1$ since $36=100100_2$)
Prove that $\alpha^3+x\alpha+1=0$.
2009 Today's Calculation Of Integral, 442
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$
2005 Today's Calculation Of Integral, 54
evaluate
\[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]
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 Kazakhstan National Olympiad, 6
Is there exist four points on plane, such that distance between any two of them is integer odd number?
May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:
2002 APMO, 1
Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
\[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \]
Prove that
\[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \]
where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?
2008 Harvard-MIT Mathematics Tournament, 19
Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.
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.
1967 IMO Longlists, 3
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$
2016 NIMO Problems, 3
Let $f$ be the quadratic function with leading coefficient $1$ whose graph is tangent to that of the lines $y=-5x+6$ and $y=x-1$. The sum of the coefficients of $f$ is $\tfrac pq$, where $p$ and $q$ are positive relatively prime integers. Find $100p + q$.
[i]Proposed by David Altizio[/i]
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$
2021 The Chinese Mathematics Competition, Problem 5
Let $D=\{ (x,y)|x^2+y^2\le \pi \}$. Find $\iint\limits_D(sin x^2cosx^2+x\sqrt{x^2+y^2})dxdy$.
2025 Romania National Olympiad, 1
Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]
2006 Harvard-MIT Mathematics Tournament, 4
Compute $\displaystyle\sum_{k=1}^\infty \dfrac{k^4}{k!}$.
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 $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
2008 Mathcenter Contest, 6
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.