Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$. [list = a] [*] Show that $\overline{AA_1}$ is a diameter of $\gamma$. [*] Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$. [/list]

Determine all functions $f : \mathbb R^+ \to \mathbb R$ that satisfy the equation $$f(xy) = f(x)f(y)f(x+y)$$ for all positive real numbers $x$ and $y$.

The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.

For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color? [i]Proposed by Eugene Chen [/i]

Positive real numbers $x,y$ satisfy $$\left \lfloor xy \right \rfloor - \lfloor x \rfloor \lfloor y \rfloor = 8.$$ Find the sum of all possible values of the quantity $\left \lfloor 2xy \right \rfloor - \lfloor 2x \rfloor \lfloor y \rfloor.$

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Show that for $n>1$ and any positive real numbers $k,x_{1},x_{2},...,x_{n}$ then \[\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}\] Where $f(x)=k^x$. When does equality hold.

[b]p1.[/b] We say that six positive integers form a magic triangle if they are arranged in a triangular array as in the figure below in such a way that each number in the top two rows is equal to the sum of its two neighbors in the row directly below it. The triangle shown is magic because $4 = 1 + 3$, $5 = 3 + 2$, and $9 = 4 + 5$. $$9$$ $$4\,\,\,\,5$$ $$1\,\,\,\,3\,\,\,\,2$$ (a) Find a magic triangle such that the numbers at the three corners are $10$, $20$, and $2010$, with $2010$ at the top. (b) Find a magic triangle such that the numbers at the three corners are $20$, $201$, and $2010$, with $2010$ at the top, or prove that no such triangle exists. [b]p2.[/b] (a) The equalities $\frac12+\frac13+\frac16= 1$ and $\frac12+\frac13+\frac17+\frac{1}{42}= 1$ express $1$ as a sum of the reciprocals of three (respectively four) distinct positive integers. Find five positive integers $a < b < c <d < e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1.$$ (b) Prove that for any integer $m \ge 3$, there exist $m$ positive integers $d_1 < d_2 <... < d_m$ such that $$\frac{1}{d_1}+\frac{1}{d_2}+ ... +\frac{1}{d_m}= 1.$$ [b]p3.[/b] Suppose that $P(x) = a_nx^n +... + a_1x + a_0$ is a polynomial of degree n with real coefficients. Say that the real number $b$ is a balance point of $P$ if for every pair of real numbers $a$ and $c$ such that $b$ is the average of $a$ and $c$, we have that $P(b)$ is the average of $P(a)$ and $P(c)$. Assume that $P$ has two distinct balance points. Prove that $n$ is at most $1$, i.e., that $P$ is a linear function. [b]p4.[/b] A roller coaster at an amusement park has a train consisting of $30$ cars, each seating two people next to each other. $60$ math students want to take as many rides as they can, but are told that there are two rules that cannot be broken. First, all $60$ students must ride each time, and second, no two students are ever allowed to sit next to each other more than once. What is the maximal number of roller coaster rides that these students can take? Justify your answer. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that the lengths of all four sides and the two diagonals of $ABCD$ are rational numbers. If the two diagonals $AC$ and $BD$ intersect at a point $M$, prove that the length of $AM$ is also a rational number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

You are at one vertex of a equilateral triangle with side length $ 1 $. All of the edges of the equilateral triangle will reflect the laser beam perfectly (angle of incidence is equal to angle of reflection). Given that the laser beam bounces off exactly $ 137 $ edges and returns to the original vertex without touching any other vertices, let $ M $ be the maximum possible distance the beam could have traveled, and $ m $ be the minimum possible distance the beam could have traveled. Find $ M^2 - m^2 $.

Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties: $\bullet$ [b]Ascension:[/b]If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable. $\bullet$ [b]Symmetry:[/b] If all of the members change their vote, the final decision will change too. Prove that the council uses a weighted decision-making method ; that is , nonnegative weights $\omega _1 , \omega _2 , \cdots ,\omega _5$ can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest. Remark. The statement isn't true at all if you replace $5$ with arbitrary $n$ . In fact , finding a counter example for $n=6$ , was appeared in the same year's [url=https://artofproblemsolving.com/community/c6h1459567p8417532]Iran MO 2nd round P6[/url]

$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$

On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds? [i]Proposed by Nathan Xiong[/i]

(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.$$ Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the [i]nonhomogeneous[/i] differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1.$ Find integers $a,b,c$ such that $(f(-2)-a)(f(2)-b)=c.$

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]