This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 1687

1986 National High School Mathematics League, 3
Tags: calculus , integration
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it integral point. Please color all intengral points in white, red and black, satisfying: (1) Points in every color appear on infinitely many lines that are parallel to $x$-axis. (2) For any white point $A$, red point $B$, black point $C$, we can find another red point $D$, such that $ABCD$ is a parallelogram.

2024 CMIMC Integration Bee, 9
Tags: calculus , integration
\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 Contests, 522
Tags: calculus , integration , limit , function , geometry , derivative , logarithm
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2013 Today's Calculation Of Integral, 899
Tags: calculus , integration , limit , calculus computations
Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2004 Vietnam Team Selection Test, 2
Tags: function , calculus , integration , search , algebra
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 $x, y \in \mathbb{R}$.

2011 Today's Calculation Of Integral, 676
Tags: calculus , integration , trigonometry , calculus computations
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$. Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$. [i]2011 Tokyo University of Science entrance exam/Management[/i]

2011 Today's Calculation Of Integral, 746
Tags: calculus , integration , floor function , inequalities , calculus computations
Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2007 Today's Calculation Of Integral, 223
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

2001 USA Team Selection Test, 2
Tags: integration , function , calculus , algebra , binomial theorem , combinatorics
Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2007 Today's Calculation Of Integral, 185
Tags: calculus , integration , trigonometry , calculus computations
Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

2011 Today's Calculation Of Integral, 716
Tags: calculus , integration , calculus computations , logarithm
Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2013 Today's Calculation Of Integral, 867
Tags: calculus , integration , quadratic , function , derivative , algebra , linear equation
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

PEN Q Problems, 2
Tags: algebra , calculus , integration , pigeonhole principle , polynomial
Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

1997 China Team Selection Test, 3
Tags: calculus , integration , algebra
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

2023 CMIMC Integration Bee, 6
Tags: calculus , integration
\[\int_0^2 e^x(x^4+8x^3+18x^2+16x+5)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

1995 Putnam, 2
Tags: ellipse , trigonometry , integration , calculus , function , conic
An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

1998 Vietnam National Olympiad, 1
Tags: integration , limit , logarithm , real analysis
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.

2006 VJIMC, Problem 4
Tags: calculus , integration
Let $f:[0,\infty)\to\mathbb R$ ba a strictly convex continuous function such that $$\lim_{x\to+\infty}\frac{f(x)}x=+\infty.$$Prove that the improper integral $\int^{+\infty}_0\sin(f(x))\text dx$ is convergent but not absolutely convergent.

2007 Harvard-MIT Mathematics Tournament, 10
Tags: calculus , integration
Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]

2007 Today's Calculation Of Integral, 226
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx$ [color=darkblue]Virgil Nicula have already posted the integral[/color] :oops:

Today's calculation of integrals, 859
Tags: calculus , integration , geometry , derivative , calculus computations
In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2008 Alexandru Myller, 3
Tags: function , real analysis , inequalities , calculus , integration , integral
Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

2010 Today's Calculation Of Integral, 577
Tags: calculus , integration , inequalities , calculus computations
Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

2013 India Regional Mathematical Olympiad, 6
Tags: calculus , integration , arithmetic sequence , number theory
Let $n \ge 4$ be a natural number. Let $A_1A_2 \cdots A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 < \cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.

2011 Today's Calculation Of Integral, 745
Tags: calculus , integration , trigonometry , calculus computations
When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$