A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$ $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

$a,b$ are real numbers (neither is $0$).Given two conditions: A: $a>0$. B: $a>b$ and $a^{-1}>b^{-1}$. Then, which one of the followings are true? $(\text{A})$A is sufficient but unnecessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient and necessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Somebody placed digits $1,2,3, \ldots , 9$ around the circumference of a circle in an arbitrary order. Reading clockwise three consecutive digits you get a $3$-digit whole number. There are nine such $3$-digit numbers altogether. Find their sum.

Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$. A configuration of $mn$ rectangles is called [i]happy[/i] if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd. [i]Proposed by Serena An and Claire Zhang[/i]

Let $K$ be an equilateral triangle in the plane. Prove that for every $p>0$ there exists an $\varepsilon>0$ with the following property: If $n$ is a positive integer, and $T_1,\ldots,T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$ \sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon, $$ then $$ \sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p. $$

Prove or disprove the following hypotheses. a) For all $k \geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k.$ b) For all $k\geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]

[asy] //rectangles above problem statement size(15cm); for(int i=0;i<5;++i){ draw((6*i-14,-1.2)--(6*i-14,1.2)--(6*i-10,1.2)--(6*i-10,-1.2)--cycle); } label("$A$", (-12,2.25)); label("$B$", (-6,2.25)); label("$C$", (0,2.25)); label("$D$", (6,2.25)); label("$E$", (12,2.25)); //top numbers label("$1$", (-12,1.25),dir(-90)); label("$0$", (-6,1.25),dir(-90)); label("$8$", (0,1.25),dir(-90)); label("$5$", (6,1.25),dir(-90)); label("$2$", (12,1.25),dir(-90)); //bottom numbers label("$9$", (-12,-1.25),dir(90)); label("$6$", (-6,-1.25),dir(90)); label("$2$", (0,-1.25),dir(90)); label("$8$", (6,-1.25),dir(90)); label("$0$", (12,-1.25),dir(90)); //left numbers label("$4$", (-14,0),dir(0)); label("$1$", (-8,0),dir(0)); label("$3$", (-2,0),dir(0)); label("$7$", (4,0),dir(0)); label("$9$", (10,0),dir(0)); //right numbers label("$6$", (-10,0),dir(180)); label("$3$", (-4,0),dir(180)); label("$5$", (2,0),dir(180)); label("$4$", (8,0),dir(180)); label("$7$", (14,0),dir(180)); [/asy] Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal. [asy] //diagram below problem statement size(7cm); for(int i=-3;i<=1;i+=2){ for(int j=-1;j<=0;++j){ if(i==1 && j==-1) continue; draw((i,j)--(i+2,j)--(i+2,j-1)--(i,j-1)--cycle); }} label("$I$",(-2,-0.5)); label("$II$",(0,-0.5)); label("$III$",(2,-0.5)); label("$IV$",(-2,-1.5)); label("$V$",(0,-1.5)); [/asy] Which of the rectangles is in position $I$? $\textbf{(A)} \ A \qquad \textbf{(B)} \ B \qquad \textbf{(C)} \ C \qquad \textbf{(D)} \ D \qquad \textbf{(E)} \ E$

Let $a$ and $b$ be positive integers with $\gcd(a, b)=1$. Show that every integer greater than $ab-a-b$ can be expressed in the form $ax+by$, where $x, y \in \mathbb{N}_{0}$.

Determine all natural numbers $x$, $y$ and $z$, such that $x\leq y\leq z$ and \[\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right) = 3\text{.}\]

For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$. [i]Proposed by Harry Kim[/i]

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$, $$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$

Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees.

At Durant University, an A grade corresponds to raw scores between $90$ and $100$, and a B grade corresponds to raw scores between $80$ and $90$. Travis has $3$ equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a $90\%$ chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.

In an acute angled triangle the feet of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .

In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. Show that $L_{2n+1}+(-1)^{n+1}(n \geq 1)$ can be written as a product of three (not necessarily distinct) Fibonacci numbers.

Quadrilateral $ABCD (\overline{AD} < \overline{BC})$ is inscribed in a circle, and $H(\neq A, B)$ is a point on segment $AB.$ The circumcircle of triangle $BCH$ meets $BD$ at $E(\neq B)$ and line $HE$ meets $AD$ at $F$. The circle passes through $C$ and tangent to line $BD$ at $E$ meets $EF$ at $G(\neq E).$ Prove that $\angle DFG = \angle FCG.$

$a,b$ are natural numbers such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ are integers.Let $d=GCD(a;b)$.Prove that $d^2\le a+b$

A chord $AB$ is drawn in a circle. On its extensions beyond points $A$ and $B$, points $P$ and $Q$ respectively are taken such that $AP = BQ$. Through $P$ and $Q$ two tangents to the circle are drawn, intersecting at point $M$. Find the locus of points $M$ ($P$ and $Q$ move along a straight line and for any $P$ and $Q$ all possible pairs of tangents are taken, which determine four points from the desired locus of points) .

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

Rank the [i]magnitudes[/i] of the [i]distance[/i] traveled during the ten second interval. (A) $\text{I} > \text{II} > \text{III}$ (B) $\text{II} > \text{I} > \text{III}$ (C) $\text{III} > \text{II} > \text{I}$ (D) $\text{I} > \text{II = III}$ (E) $\text{I = II = III}$