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

Tags were heavily modified to better represent problems.

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Found problems: 1687

2009 Today's Calculation Of Integral, 410

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.

2009 All-Russian Olympiad, 4

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

1962 Miklós Schweitzer, 7

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

Oliforum Contest II 2009, 3

Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$. [i](Paolo Leonetti)[/i]

2010 Today's Calculation Of Integral, 580

Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$. Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

2014 Putnam, 6

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

2007 F = Ma, 31

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. Find the ratio $L/d$. $ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

2010 Today's Calculation Of Integral, 544

(1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$. (2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}$, then we have $ f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1)$, thus we have $ f(x)\equal{}\boxed{\ C\ }$.

2014 Online Math Open Problems, 30

Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

2001 Moldova National Olympiad, Problem 7

Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$

2015 BMT Spring, 15

Compute $$\int_{1/2}^{2} \frac{x^2 + 1}{x^2(x^{2015} + 1)} dx.$$

2009 Purple Comet Problems, 8

Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$.

2011 Today's Calculation Of Integral, 715

Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.

2011 Today's Calculation Of Integral, 746

Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2013 Princeton University Math Competition, 1

Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.

2005 ISI B.Math Entrance Exam, 6

Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

2006 AMC 12/AHSME, 20

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

2016 District Olympiad, 4

Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence $$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$ is convergent and calculate its limit.

2009 Today's Calculation Of Integral, 507

Evaluate \[ \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]

1974 Miklós Schweitzer, 7

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

2010 Romania Team Selection Test, 4

Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$. [i]Marius Cavachi[/i]

2005 Alexandru Myller, 3

Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$. [i]Gabriel Marsanu, Andrei Nedelcu[/i]

1970 IMO Longlists, 52

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2013 Today's Calculation Of Integral, 861

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

1951 AMC 12/AHSME, 15

The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$