Found problems: 2215
2010 Tuymaada Olympiad, 1
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
2011 Today's Calculation Of Integral, 695
For a positive integer $n$, let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$.
(3) Find the limit $\lim_{n\to\infty} T_n.$
2013 Romania National Olympiad, 4
a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$.
b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that
$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$
2022 JHMT HS, 8
Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?
2013 Singapore MO Open, 5
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.
2013 AMC 12/AHSME, 13
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
2016 Danube Mathematical Olympiad, 3
3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1.
a) Show that there exists a constant c ≥ 1/2 so that
$\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$,
where $a_0 = 0$.
b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.
PEN S Problems, 38
The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.
2011 Today's Calculation Of Integral, 676
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]
2019 Korea USCM, 3
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
1964 Miklós Schweitzer, 8
Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.
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]
1998 Harvard-MIT Mathematics Tournament, 3
Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.
2011 Today's Calculation Of Integral, 735
Evaluate the following definite integrals:
(a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$
(b) $\int_0^{\frac 13} xe^{3x}\ dx$
(c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$
(d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$
2009 Jozsef Wildt International Math Competition, W. 7
If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$
1997 China Team Selection Test, 3
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 $\forall$ $n \geq 1$.
2013 Today's Calculation Of Integral, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
2009 Today's Calculation Of Integral, 459
Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.
2008 Harvard-MIT Mathematics Tournament, 1
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?
2012 Today's Calculation Of Integral, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
2014 Tuymaada Olympiad, 8
Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that
\[ g(a, b, c)\ge \sqrt{2abc}\]
[i](M. Ivanov)[/i]
[hide="Remarks (containing spoilers!)"]
1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$.
2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$.
[/hide]
2023 AIME, 6
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(2.5cm);
draw((0,0)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--cycle);
draw((0,1)--(1,1)--(1,0), dotted);
[/asy]
2001 Romania National Olympiad, 3
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.
b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
2007 F = Ma, 6
At time $t = 0$ a drag racer starts from rest at the origin and moves along a straight line with velocity given by $v = 5t^2$, where $v$ is in $\text{m/s}$ and $t$ in $\text{s}$. The expression for the displacement of the car from $t = 0$ to time $t$ is
$ \textbf{(A)}\ 5t^3 \qquad\textbf{(B)}\ 5t^3/3\qquad\textbf{(C)}\ 10t \qquad\textbf{(D)}\ 15t^2 \qquad\textbf{(E)}\ 5t/2 $
1981 Putnam, B6
Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with
$$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$
for all polygons $P$, where the double integral is taken over $H(P).$