How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

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 $m-n$.

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$ $ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $

Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov

In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis. Note you may not use double integral.

Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

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

If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

Evaluate \[\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}\]

Determine the functions $f:(0,\pi)\to\mathbb R$ which satisfy $$f'(x)=\frac{\cos2020x}{\sin x}$$ for any real $x\in(0,\pi)$. [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

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.

About the equation $ ax^2 \minus{} 2x\sqrt {2} \plus{} c \equal{} 0$, with $ a$ and $ c$ real constants, we are told that the discriminant is zero. The roots are necessarily: $ \textbf{(A)}\ \text{equal and integral} \qquad\textbf{(B)}\ \text{equal and rational} \qquad\textbf{(C)}\ \text{equal and real}$ $ \textbf{(D)}\ \text{equal and irrational} \qquad\textbf{(E)}\ \text{equal and imaginary}$