Let $p>1,\frac1p+\frac1q=1$ and $r>1$. If $u(x,y),v(x,y)>0$, and $f(x,y),g(x,y)$ are continuous functions on $[a,b]\times[c,d]$, then prove $$\left(\frac{\left(\int^b_a\int^d_c(f(x,y)+g(x,y))^rdxdy\right)^{1/r}}{(u(x,y)+v(x,y))^{1/q}}\right)^p\le\left(\frac{\left(\int^b_a\int^d_cf(x,y)^rdxdy\right)^{1/r}}{u(x,y)^{1/q}}\right)^p+\left(\frac{\left(\int^b_a\int^d_cg(x,y)^rdxdy\right)^{1/r}}{v(x,y)^{1/q}}\right)^p,$$ with equality if and only if either $$\left(\lVert f(x,y)\rVert^r_r,\lVert g(x,y)\rVert^r_r\right)=\alpha\left(\lVert u(x,y)\rVert^r_r,\lVert v(x,y)\rVert^r_r\right)$$ for some $\alpha>0$ or $\lVert f(x,y)\rVert^r_r=\lVert g(x,y)\rVert^r_r=0$. [i]Proposed by Chang-Jian Zhao[/i]

\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Suppose $p''(x) = 4x^2 + 4x + 2$ where $$p(x) = a_0 + a_1(x - 1) + a_2(x -2)^2 + a_3(x- 3)^4 + a_4(x-4)^4.$$ We have $p'(-3) = -24$ and $p(x)$ has the unique property that the sum of the third powers of the roots of $p(x)$ is equal to the sum of the fourth powers of the roots of $p(x)$ . Find $a_0$.

[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

$$\int_0^{20\pi}|x\sin(x)|\,dx$$ [i]Proposed by Connor Gordon[/i]

Find the sum of the series $ x\plus{}\frac{x^3}{1\cdot 3}\plus{}\frac{x^5}{1\cdot 3\cdot 5}\plus{}\cdots\plus{}\frac{x^{2n\plus{}1}}{1\cdot 3\cdot 5\cdot \cdots \cdot (2n\plus{}1)}\plus{}\cdots$

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 $

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear. [i]Valentin Vornicu[/i]

Suppose that $f$ is a differentiable function such that $f(0) = 20$ and $|f'(x)| \leq 4$ for all real numbers $x$. Let $a$ and $b$ be real numbers such that [i]every[/i] such function $f$ satisfies $a \leq f(22) \leq b$. Find the smallest possible value of $|a| + |b|$.

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that $$ xf(x)\ge \int_0^x f(t)dt , $$ for all real numbers $ x. $ Prove that [b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $ [b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant. [i]Mihai Piticari[/i]

Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.

Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

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 $n$.

Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

A certain type of Bessel function has the form $I(x) = \frac{1}{\pi} \int_0^{\pi}e^{x \cos \theta} d\theta$ for all real $x$. Evaluate $\int_0^{\infty} x I(2x) e^{-x^2}dx$.

Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$

Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity \[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$. [i]Kiran Kedlaya and Peter Shor.[/i]