Prove: there are polynomials $S_1, S_2, \ldots$ in the variables $x_1, x_2, \ldots,y_1, y_2,\ldots$ with integer coefficients satisfying, for every integer $n \ge 1$, $$\sum_{d \mid n} d \cdot S_d ^{n/d}=\sum_{d \mid n} d \cdot (x_d ^{n/d}+y_d ^{n/d}) \quad (*)$$ Here, the sums run through the positive divisors $d$ of $n$. For example, the first two polynomials are $S_1 = x_1 + y_1$ and $S_2 = x_2 + y_2 - x_1y_1$, which verify identity $(*)$ for $n = 2$: $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \cdot(x_2 + y_2)$.

Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.

Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

Anna writes a sequence of integers starting with the number 12. Each subsequent integer she writes is chosen randomly with equal chance from among the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops. One such sequence is \[ 12, 6, 6, 3, 3, 3, 1. \] What is the expected value of the number of terms in Anna’s sequence?

Given real positive a,b , find the larget real c such that $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$ for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.

Several identical cars are standing at a red light on a one-lane road, one behind the other, with negligible (and equal) distance between adjacent cars. When the green light comes up, the first car takes off to the right with constant acceleration. The driver in the second car reacts and does the same 0.2 s later. The third driver starts moving 0.2 s after the second one and so on. All cars accelerate until they reach the speed limit of 45 km/hr, after which they move to the right at a constant speed. Consider the following patterns of cars. Just before the first car starts accelerating to the right, the car pattern will qualitatively look like the pattern in I. After that, the pattern will qualitatively evolve according to which of the following? $\textbf{(A)}\text{first I, then II, and then III}$ $\textbf{(B)}\text{first I, then II, and then IV}$ $\textbf{(C)}\text{first I, and then IV, with neither II nor III as an intermediate stage}$ $\textbf{(D)}\text{first I, and then II}$ $\textbf{(E)}\text{first I, and then III}$

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$, $DF=8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m,n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.

Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively. $P\in AB,Q\in AC$ satisfy $NP\perp AB,MQ\perp AC$ a) Prove that EF is the tangent line of (APQ) b) Let T be the tangency point of (APQ) with EF,.DT $\cap$ MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont

What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$? $ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 61 \qquad\textbf{(C)}\ 41 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 01 $

Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$. [i](5 points)[/i]

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$.

Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

Let $a>1$ be an integer which is not divisible by four. Prove that there are infinitely many primes $p$ of the form $4k-1$ such that $p | a^d-1$ for some $d<\frac{p-1}{2}$