Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.

Which species has the largest $\text{F-A-F}$ bond angle where $\text{A}$ is the central atom? $ \textbf{(A) }\ce{BF3} \qquad\textbf{(B) }\ce{CF4} \qquad\textbf{(C) }\ce{NF3}\qquad\textbf{(D) }\ce{OF2}\qquad$

Find all positive integers n with the following property: For every positive divisor $d$ of $n$, $d+1$ divides $n+1$.

In the following diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ so that the area of the square is $m +\sqrt{n}$. Find $m + n$. [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; filldraw((-0.707106781,0.707106781)--(-0.707106781,-1)--(1,-1)--(1,0.707106781)--cycle,gray, linewidth(1.4)); draw(circle((0,0),1), linewidth(1.4)); [/asy]

Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?

The graph shows velocity as a function of time for a car. What was the acceleration at time = $90$ seconds? [asy] size(275); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2)); draw((0,3)--(6,3)); draw((0,4)--(6,4)); draw((0,0)--(0,4)); draw((1,0)--(1,4)); draw((2,0)--(2,4)); draw((3,0)--(3,4)); draw((4,0)--(4,4)); draw((5,0)--(5,4)); draw((6,0)--(6,4)); label("$0$",(0,0),S); label("$30$",(1,0),S); label("$60$",(2,0),S); label("$90$",(3,0),S); label("$120$",(4,0),S); label("$150$",(5,0),S); label("$180$",(6,0),S); label("$0$",(0,0),W); label("$10$",(0,1),W); label("$20$",(0,2),W); label("$30$",(0,3),W); label("$40$",(0,4),W); draw((0,0.6)--(0.1,0.55)--(0.8,0.55)--(1.2,0.65)--(1.9,1)--(2.2,1.2)--(3,2)--(4,3)--(4.45,3.4)--(4.5,3.5)--(4.75,3.7)--(5,3.7)--(5.5,3.45)--(6,3)); label("Time (s)", (7.5,0),S); label("Velocity (m/s)",(-1,3),W); [/asy] $ \textbf{(A)}\ 0.2\text{ m/s}^2\qquad\textbf{(B)}\ 0.33\text{ m/s}^2\qquad\textbf{(C)}\ 1.0\text{ m/s}^2\qquad\textbf{(D)}\ 9.8\text{ m/s}^2\qquad\textbf{(E)}\ 30\text{ m/s}^2 $

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following. For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?

Let $\triangle ABC$ be inscribed in circle $O$ with $\angle ABC = 36^\circ$. $D$ and $E$ are on the circle such that $\overline{AD}$ and $\overline{CE}$ are diameters of circle $O$. List all possible positive values of $\angle DBE$ in degrees in order from least to greatest. [i]Proposed by Ambrose Yang[/i]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Let $AG, CH$ be the angle bisectors of a triangle $ABC$. It is known that one of the intersections of the circles of triangles $ABG$ and $ACH$ lies on the side $BC$. Prove that the angle $BAC$ is $60 ^o$

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)