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

1985 AIME Problems, 10
Tags: floor function , calculus , integration , function
How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

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

2012 Today's Calculation Of Integral, 783
Tags: calculus , integration , calculus computations
Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$. (1) Find $a_n$. (2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$. Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$. Last Edited

2009 Today's Calculation Of Integral, 475
Tags: calculus , integration , calculus computations
For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$

2007 Putnam, 6
Tags: integration , probability , analytic geometry , logarithm
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$ \[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]

2011 Today's Calculation Of Integral, 733
Tags: calculus , integration , limit , calculus computations
Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

2005 Alexandru Myller, 1
Tags: real analysis , function , integration
Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$. [i]Eugen Paltanea[/i]

2020 Jozsef Wildt International Math Competition, W50
Tags: calculus , integration , inequalities
Let $f:[0,1]\to\mathbb R$ be a differentiable function, while $f'$ is continuous on $[0,1]$ and $|f'(x)|\le1$, $(\forall)x\in[0,1]$. If $$2\left|\int^1_0f(x)dx\right|\le1$$ Show that: $$(n+2)\left|\int^1_0x^nf(x)dx\right|\le1,~(\forall)x\ge1$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

2007 IberoAmerican Olympiad For University Students, 3
Tags: calculus , integration , inequalities , function , real analysis , integral inequality , logarithm
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

2007 QEDMO 5th, 5
Tags: induction , algebra , function , domain , calculus , integration , polynomial
Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

2012 Turkey Junior National Olympiad, 1
Tags: quadratic , calculus , integration , diophantine equation , number theory
Let $x, y$ be integers and $p$ be a prime for which \[ x^2-3xy+p^2y^2=12p \] Find all triples $(x,y,p)$.

2006 Putnam, A1
Tags: analytic geometry , calculus , integration , trigonometry , geometry , projective geometry
Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

2010 Today's Calculation Of Integral, 639
Tags: calculus , integration , calculus computations
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

1951 Miklós Schweitzer, 5
Tags: integration , probability and stats
In a lake there are several sorts of fish, in the following distribution: $ 18\%$ catfish, $ 2\%$ sturgeon and $ 80\%$ other. Of a catch of ten fishes, let $ x$ denote the number of the catfish and $ y$ that of the sturgeons. Find the expectation of $ \frac {x}{y \plus{} 1}$

1979 AMC 12/AHSME, 11
Tags: calculus , integration
Find a positive integral solution to the equation \[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\] $\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$

1983 Putnam, A6
Tags: limit , calculus , integration , real analysis
Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

1991 Arnold's Trivium, 21
Tags: calculus , derivative , integration
Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

2022 CMIMC Integration Bee, 13
Tags: calculus , integration
\[\int_{-\infty}^\infty e^{-x^2-4/x^2}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Today's Calculation Of Integral, 34
Tags: calculus , integration , probability , calculus computations
Let $p$ be a constant number such that $0<p<1$. Evaluate \[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]

2014 Contests, 2
Tags: calculus , integration , number theory
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

2009 Today's Calculation Of Integral, 401
Tags: calculus , integration , trigonometry , symmetry , limit , calculus computations
For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

2011 Today's Calculation Of Integral, 684
Tags: calculus , integration , function , parabola , calculus computations , conic , geometry
On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]

2007 Today's Calculation Of Integral, 219
Tags: calculus , integration , limit , algebra , function , domain , logarithm
Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$. Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.

2010 Today's Calculation Of Integral, 665
Tags: calculus , integration , limit , trigonometry , calculus computations
Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

2011 Today's Calculation Of Integral, 763
Tags: calculus , integration , function , calculus computations , logarithm
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$