Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that \[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]

Let $p>5$ be a prime number. For any integer $x$, define \[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\] Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.

Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.

Click for solution The case $k=2.5$ is trivial. There is counterexample! Let $k=3$. Without loss of generality let $P_1 \equiv (-3,0)$ and $P_n \equiv (3,0)$. Let $M$ be $S$ under translation by the vector $-\frac{1}{3} . \vec{P_1P_n}$. Let $N$ be $S$ under translation by the vector $\frac{1}{3} . \vec{P_1P_n}$. Suppose that we can't find $A$ and $B$ such that $k=3$ and $AB||P_1P_n$. This means that $M\cap S=\emptyset$ and $N\cap S=\emptyset$ Now we have three broken lines $M,S$ and $N$. Let's continue $S$ in both directions so that we divide the plane into two parts $\mu$ and $\nu$, and M is in the part $\mu$ and N is in the part $\nu$. Lets look at the $Ox$ axis. Each point of it is either in $\mu$ or $\nu$ or in the border $S$. The point $-3$ is on the border, so $-1$ is in $\nu$. $3$ is on the border, so $1$ is in $\mu$. So there exist $-1\leq B_1 \leq 1$ so that $B_1$ is on the border. So $-3\leq B_1 -2\leq -1$ and $M_1=B_1 -2$ is in $\mu$. So there exist $M_1\leq B_2 \leq N$ with $B_2$ on the border. Then $N_2= B_2 +2$ is between $B_1$ and $1$. There exist $B_3$ between $N_2$ and $1$ on the border and we build infinite sequence $B_{2k+1}$ of points on the border and between every two there is point not on the border. That means that the broken line S intersects the $Ox$ axis infinite number of times- \textbf{CONTRADICTION}

Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.

Consider the family of nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios \[ \frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE} \] is constant.

Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.

These problems are copyright $\copyright$ Mathematical Association of America.