If $ab\ne0$ and $|a|\ne|b|$ the number of distinct values of $x$ satisfying the equation \[\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}\] is: $\text{(A)}\ \text{zero}\qquad \text{(B)}\ \text{one}\qquad \text{(C)}\ \text{two}\qquad \text{(D)}\ \text{three}\qquad \text{(E)}\ \text{four}$

Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25% faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.

Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\neq0$. Alan must respond with the value $Q(s,t)$. Allen's goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer, $g$, such that Allen can always achieve this goal making no more than $g$ queries. [i]Linus Tang[/i]

In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

Consider a triangle $ABC$ in which $AB<BC<CA$. The excircles touch the sides $BC, CA,$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. A circle is drawn through the points $A_1, B_1$ and $C_1$ which intersects the sides $BC, CA$ and $AB$ for the second time at the points $A_2, B_2$ and $C_2$ respectively. On which side of the triangle can lie the largest of the segments $A_1A_2, B_1B_2$ and $C_1C_2$? [i]Proposed by I. Weinstein[/i]

Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribe an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is parallel to the major axis. Compute the area of $\vartriangle ABC$.

Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$

Let $a,b$ and $c$ be real numbers with $b,c >0.$ Prove that if $ a<b \ ( a>b),$ then \[\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab) \] And then prove that $\frac{a+c}{b+c}$ is between $1$ and $\frac ab.$

For numbers $a,b \in \mathbb{R}$ we consider the sets: $$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$ Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$: $$ P(r) \in A \iff Q(r) \in B$$

Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$.

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$ [asy] size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy] $\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Suppose that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$. Prove that $\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab$.

Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt 5$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt 5$. What is $|AD|$? $ \textbf{(A)}\ 10\sqrt 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 4\sqrt{15} \qquad\textbf{(D)}\ 7\sqrt 5 \qquad\textbf{(E)}\ \text{None of above} $

Let $A_1,\ldots,A_m$ be three-element subsets of an $n$-element set $X$ such that $|A_i\cup A_j|\le1$ whenever $i\ne j$. Prove that there exists a subset $A$ of $X$ with $|A|\ge2\sqrt n$ such that it does not contain any of the $A_i$.

The points $P$ and $Q$ are placed in the interior of the triangle $\Delta ABC$ such that $m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)$ and similarly for the other $2$ vertices($P$ and $Q$ are isogonal conjugates). Let $P_{A}$ and $Q_{A}$ be the intersection points of $AP$ and $AQ$ with the circumcircle of $CPB$, respectively $CQB$. Similarly the pairs of points $(P_{B},Q_{B})$ and $(P_{C},Q_{C})$ are defined. Let $PQ_{A}\cap QP_{A}=\{M_{A}\}$, $PQ_{B}\cap QP_{B}=\{M_{B}\}$, $PQ_{C}\cap QP_{C}=\{M_{C}\}$. Prove the following statements: $1.$ Lines $AM_{A}$, $BM_{B}$, $CM_{C}$ concur. $2. $ $M_{A}\in BC$, $M_{B}\in CA$, $M_{C}\in AB$

in a right-angled triangle $ABC$ with $\angle C=90$,$a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$;$a,c$ respectively,with radii $r,t$.find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.