Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$. We regard $k$ as fixed. Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$. For what ratio $XB/BY$ will the segment $XY$ have minimal length?

Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$ [i]Proposed by Ada Tsui[/i]

Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$. (houkai)

Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$. Determine the smallest value of $M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}$.

Among the $5$-sided polygons, as many vertices as possible collinear , that is, belonging to a single line, is three, as shown below. What is the largest number of collinear vertices a $12$-sided polygon can have? [img]https://cdn.artofproblemsolving.com/attachments/1/1/53d419efa4fc4110730a857ae6988fc923eb13.png[/img] Attention: In addition to drawing a $12$-sided polygon with the maximum number of vertices collinear , remember to show that there is no other $12$-sided polygon with more vertices collinear than this one.

Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.

In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\] for all $x,y\in\mathbb{R}$.

A ring $ A $ has property [i](P),[/i] if $ A $ is finite and there exists $ (\{ 0\}\neq R,+)\le (A,+) $ such that $ (U(A),\cdot )\cong (R,+) . $ Show that: [b]a)[/b] If a ring has property [i](P),[/i] then, the number of its elements is even. [b]b)[/b] There are infinitely many rings of distinct order that have property [i](P).[/i]

Bridget bakes $48$ loaves of bread for her bakery. She sells half of them in the morning for $\$2.50$ each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs $\$0.75$ for her to make. In dollars, what is her profit for the day? ${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 52$

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

Find the sum of all even positive integers less than $233$ not divisible by $10$.

Suppose that $p(n)$ is the number of partitions of a natural number $n$. Prove that there exists $c>0$ such that $P(n)\ge n^{c \cdot \log n}$. [i]proposed by Mohammad Mansouri[/i]

It is well-known that if $\frac{p}{q}=\frac{r}{s}$, both of the expressions are also equal to $\frac{p-r}{q-s}$. Now we write the equality $$\frac{3x-b}{3x-5b}=\frac{3a-4b}{3a-8b}.$$ The previous property shows that both fractions should be equal to $$\frac{3x-b-3a+4b}{3x-5b-3a+8b}=\frac{3x-3a+3b}{3x-3a+3b}=1.$$ However, the initial fractions given may not be equal to $1$. Explain what is going on.

Let $ABCD$ be a cyclic quadrilateral so that arcs $AB$ and $BC$ are equal. Given that $AD = 6, BD = 4$ and $CD = 1$, compute $AB$.

In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?

Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?

The bar graph shows the results of a survey on color preferences. What percent preferred blue? [asy] for (int a = 1; a <= 6; ++a) { draw((-1.5,4*a)--(1.5,4*a)); } draw((0,28)--(0,0)--(32,0)); draw((3,0)--(3,20)--(6,20)--(6,0)); draw((9,0)--(9,24)--(12,24)--(12,0)); draw((15,0)--(15,16)--(18,16)--(18,0)); draw((21,0)--(21,24)--(24,24)--(24,0)); draw((27,0)--(27,16)--(30,16)--(30,0)); label("$20$",(-1.5,8),W); label("$40$",(-1.5,16),W); label("$60$",(-1.5,24),W); label("$\textbf{COLOR SURVEY}$",(16,26),N); label("$\textbf{F}$",(-6,25),W); label("$\textbf{r}$",(-6.75,22.4),W); label("$\textbf{e}$",(-6.75,19.8),W); label("$\textbf{q}$",(-6.75,17.2),W); label("$\textbf{u}$",(-6.75,15),W); label("$\textbf{e}$",(-6.75,12.4),W); label("$\textbf{n}$",(-6.75,9.8),W); label("$\textbf{c}$",(-6.75,7.2),W); label("$\textbf{y}$",(-6.75,4.6),W); label("D",(4.5,.2),N); label("E",(4.5,3),N); label("R",(4.5,5.8),N); label("E",(10.5,.2),N); label("U",(10.5,3),N); label("L",(10.5,5.8),N); label("B",(10.5,8.6),N); label("N",(16.5,.2),N); label("W",(16.5,3),N); label("O",(16.5,5.8),N); label("R",(16.5,8.6),N); label("B",(16.5,11.4),N); label("K",(22.5,.2),N); label("N",(22.5,3),N); label("I",(22.5,5.8),N); label("P",(22.5,8.6),N); label("N",(28.5,.2),N); label("E",(28.5,3),N); label("E",(28.5,5.8),N); label("R",(28.5,8.6),N); label("G",(28.5,11.4),N); [/asy] $\text{(A)}\ 20\% \qquad \text{(B)}\ 24\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 36\% \qquad \text{(E)}\ 42\% $

Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.

Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying \[f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,\] where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings.

Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.

Let $\{a_n\}_{n=0}^{\infty}$ be the sequence defined by the recurrence relation $a_{n+3}=2a_{n+2} - 23a_{n+1}+3a_n$ for all $n \ge 0,$ with initial conditions $a_0=20, a_1=0,$ and $a_2=23.$ Let $b_n=a_n^3$ for all $n \ge 0.$ There exists a unique positive integer $k$ and constants $c_0, \ldots, c_{k-1}$ with $c_0 \neq 0$ and $c_{k-1} \neq 0$ such that for all sufficiently large $n,$ we have the recurrence relation $b_{n+k} = \sum_{t=0}^{k-1} c_t b_{n+t}.$ Find $k+\sqrt{|c_{k-1}|}+\sqrt{|c_0|}.$

Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

Let $n>2$ be an integer. In a country there are $n$ cities and every two of them are connected by a direct road. Each road is assigned an integer from the set $\{1, 2,\ldots ,m\}$ (different roads may be assigned the same number). The [i]priority[/i] of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest $m$ for which it is possible that all cities have a different priority.

Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality: \[f(3x+2y)=f(x)f(y)\] for all non-negative integers $x,y$.