Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? $ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1.[/b] If the values of two angles in a triangle are $60$ and $75$ degrees respectively, what is the measure of the third angle? [b]B2.[/b] Square $ABCD$ has side length $1$. What is the area of triangle $ABC$? [b]B3 / G1.[/b] An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is $8$, what is the square’s side length? [b]B4 / G2.[/b] What is the maximum possible number of sides and diagonals of equal length in a quadrilateral? [b]B5.[/b] A square of side length $4$ is put within a circle such that all $4$ corners lie on the circle. What is the diameter of the circle? [b]B6 / G3.[/b] Patrick is rafting directly across a river $20$ meters across at a speed of $5$ m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of $12$ m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled? [b]B7 / G4.[/b] Quadrilateral $ABCD$ has side lengths $AB = 7$, $BC = 15$, $CD = 20$, and $DA = 24$. It has a diagonal length of $BD = 25$. Find the measure, in degrees, of the sum of angles $ABC$ and $ADC$. [b]B8 / G5.[/b] What is the largest $P$ such that any rectangle inscribed in an equilateral triangle of side length $1$ has a perimeter of at least $P$? [b]G6.[/b] A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$. [b]G7.[/b] Let $\vartriangle ABC$ be such that $\angle A = 105^o$, $\angle B = 45^o$, $\angle C = 30^o$. Let $M$ be the midpoint of $AC$. What is $\angle MBC$? [b]G8.[/b] Points $A$, $B$, and $C$ lie on a circle centered at $O$ with radius $10$. Let the circumcenter of $\vartriangle AOC$ be $P$. If $AB = 16$, find the minimum value of $PB$. [i]The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides. [/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$

In triangle $ABC$, let $O$ and $I_A$ be the centers of the circumcircle and the circle tangent to $AB$ and $AC$ and externally tangent to $BC$, and let $R$ and $R_A$ be their radii. Find $ \frac {I_A A \cdot I_A B \cdot I_A C}{R \cdot R_A^2} $.

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$

In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$. [i]Proposed by Evan Chen[/i]

Let $P(x)$ be a polynomial with real coefficients. Prove that not all roots of $x^3P(x)+1$ are real.

[b]7.[/b] Prove that for any odd prime number $p$, the polynomial $2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$ is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b] *This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594

An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?

Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

A solution of $\ce{KNO3}$ in water is prepared for which the following data have been obtained: \[ \text{masses of solute and solvent} \] \[ \text{molar masses of solute and solvent} \] Which of these quantitative descriptions of the solution can be determined? \[ \text{I. molarity II. molality III. density of solution} \] $ \textbf{(A) }\text{I. only}\qquad\textbf{(B) }\text{II. only}\qquad\textbf{(C) }\text{I. and II. only}\qquad\textbf{(D) }\text{I., II. and III.}\qquad$

There are $315$ marbles divided into three piles of $81, 115$ and $119$. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into $2$ equal piles. Can Ah Meng divide the marbles into $315$ piles, each with a single marble?

Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$

Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?

Given an increasing sequence of different natural numbers $a_1 < a_2 < a_3 < ... < a_n$ such that for any two distinct numbers in this sequence their sum is not divisible by $10$. It is known that $a_n = 2023$. a) Can $n$ be greater than $800$? b) What is the largest possible value of $n$? c) For the value $n$ found in question b), find the number of such sequences with $a_n = 2023$.