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: 713

Today's calculation of integrals, 864
Tags: calculus , integration , limit , inequalities , calculus computations
Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2010 Today's Calculation Of Integral, 563
Tags: integration , calculus , function , quadratic , calculus computations
Determine the pair of constant numbers $ a,\ b,\ c$ such that for a quadratic function $ f(x) \equal{} x^2 \plus{} ax \plus{} b$, the following equation is identity with respect to $ x$. \[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\] .

2010 Today's Calculation Of Integral, 587
Tags: integration , calculus , calculus computations
Evaluate $ \int_0^1 \frac{(x^2\plus{}3x)e^x\minus{}(x^2\minus{}3x)e^{\minus{}x}\plus{}2}{\sqrt{1\plus{}x(e^x\plus{}e^{\minus{}x})}}\ dx$.

2009 Today's Calculation Of Integral, 506
Tags: conic , integration , parabola , function , calculus computations , geometry , calculus
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

2010 Today's Calculation Of Integral, 523
Tags: logarithm , integration , calculus computations , calculus
Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2012 Today's Calculation Of Integral, 811
Tags: trigonometry , calculus , calculus computations , integration
Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

2009 Today's Calculation Of Integral, 409
Tags: calculus computations , calculus , integration
Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

2010 Today's Calculation Of Integral, 646
Tags: logarithm , integration , calculus , calculus computations , trigonometry
Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2012 Today's Calculation Of Integral, 830
Tags: calculus computations , calculus , logarithm , limit , function , integration
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

2011 Today's Calculation Of Integral, 744
Tags: calculus computations , inequalities , calculus , integration
Let $a,\ b$ be real numbers. If $\int_0^3 (ax-b)^2dx\leq 3$ holds, then find the values of $a,\ b$ such that $\int_0^3 (x-3)(ax-b)dx$ is minimized.

2005 Today's Calculation Of Integral, 19
Tags: logarithm , calculus , trigonometry , calculus computations , integration
Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

2010 Today's Calculation Of Integral, 621
Tags: calculus , logarithm , limit , real analysis , calculus computations , integration
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

2009 Today's Calculation Of Integral, 413
Tags: integration , calculus computations , trigonometry , calculus , derivative
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

2009 Today's Calculation Of Integral, 516
Tags: integration , calculus , trigonometry , logarithm , calculus computations
Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2010 Today's Calculation Of Integral, 591
Tags: calculus computations , calculus , inequalities , integration
Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

2007 Today's Calculation Of Integral, 236
Tags: trigonometry , integration , calculus computations , calculus
Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$. \[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\]. [i]1998 Shinsyu University entrance exam/Textile Science[/i]

2011 Today's Calculation Of Integral, 734
Tags: calculus , logarithm , calculus computations , integration
Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

Today's calculation of integrals, 848
Tags: trigonometry , calculus , logarithm , integration , calculus computations
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2009 Today's Calculation Of Integral, 495
Tags: trigonometry , calculus , integration , logarithm , calculus computations
Evaluate the following definite integrals. (1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$ (2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$ (3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$

2012 Today's Calculation Of Integral, 821
Tags: calculus , logarithm , calculus computations , integration , inequalities
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

2014 Contests, 900
Tags: calculus computations , integration , calculus , trigonometry
Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

2007 Today's Calculation Of Integral, 175
Tags: integration , logarithm , calculus , calculus computations
Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

2013 Today's Calculation Of Integral, 888
Tags: integration , calculus , analytic geometry , trigonometry , calculus computations , geometry
In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

2012 Today's Calculation Of Integral, 837
Tags: logarithm , calculus , limit , integration , calculus computations
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$ Find $\lim_{n\to\infty} f_n(1).$

2009 Today's Calculation Of Integral, 429
Tags: calculus , trigonometry , calculus computations , integration
Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.