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

2004 IMC, 2

Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have \[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \] Prove that \[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]

2010 Today's Calculation Of Integral, 605

Let $f(x)$ be a differentiable function. Find the following limit value: \[\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.\] Especially, for $f(x)=(x-\alpha)(x-\beta)$ find the limit value above. 1956 Tokyo Institute of Technology entrance exam

1994 Balkan MO, 2

Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \] [i]Greece[/i]

2023 OMpD, 4

Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$

1963 Putnam, A3

Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$

2011 Today's Calculation Of Integral, 717

Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis. Prove that : \[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]

2005 Putnam, A6

Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?

1983 Miklós Schweitzer, 12

Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$. [i]S. Csorgp, L. Horvath[/i]

1974 AMC 12/AHSME, 26

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

2009 Today's Calculation Of Integral, 519

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

1994 Cono Sur Olympiad, 2

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

2007 Today's Calculation Of Integral, 215

For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$. Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.

2009 Today's Calculation Of Integral, 492

Find the volume formed by the revolution of the region satisfying $ 0\leq y\leq (x \minus{} p)(q \minus{} x)\ (0 < p < q)$ in the coordinate plane about the $ y$ -axis. You are not allowed to use the formula: $ V \equal{} \boxed{\int_a^b 2\pi x|f(x)|\ dx\ (a < b)}$ here.

2010 Today's Calculation Of Integral, 668

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin University entrance exam/Medicine [/i]

2009 Unirea, 4

Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

2008 District Olympiad, 1

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that $$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$ Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $

2009 Today's Calculation Of Integral, 447

Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

2007 Putnam, 2

Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]

2018 Brazil Undergrad MO, 22

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

2010 Today's Calculation Of Integral, 640

Evaluate $\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.$ Own

2009 Today's Calculation Of Integral, 448

Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$

2005 Romania National Olympiad, 3

Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

2011 Today's Calculation Of Integral, 688

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

2002 AMC 12/AHSME, 24

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

2010 China Girls Math Olympiad, 8

Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$