[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$. (a) Represent $29 - 12\sqrt5$ as a prefect square. (b) Represent $10 - 6\sqrt3$ as a prefect cube. [b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions. $$x+cy = 1$$ $$cx+9y = 3$$ [b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$. (a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point. (b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation. [b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it. Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?” “Well, Miranda, could you tell me something else?” “Sure”, said Miranda, “my age is an odd number”. [b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway. [img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img] [b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? (b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin? PS. You should use hide for answers.

Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$ [img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]

Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly. (a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel. (b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear. [i]Nikolai Beluhov[/i]

Consider the set of all pairs of positive integers $(A , B)$ in which $A < B$ . Some of these pairs are to $be$ designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form $A, A + D, A + 2D$, in which $D > 0$, the associated pairs $(A, A + D )$ , $(A , A + 2D)$ and $(A + D, A + 2D)$ would include at least one pair of each colour?

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

[u]Set 8[/u] [b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends. [b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid. [b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings such that any integer $ n \ge 2$ has that (a) If $n$ is prime, then at least one digit of $n$ is not blue. (b) If $n$ is composite, then at least one digit of $n$ is not green. PS. You should use hide for answers.

Say that an integer $n \ge 2$ is [i]delicious[/i] if there exist $n$ positive integers adding up to 2014 that have distinct remainders when divided by $n$. What is the smallest delicious integer?

Let $c_1, \cdots, c_n \ (n \geq 2)$ be real numbers such that $0 \leq \sum c_i \leq n$. Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$

Positive integers $ a$, $ b$, and $ 2009$, with $ a<b<2009$, form a geometric sequence with an integer ratio. What is $ a$? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 289 \qquad \textbf{(E)}\ 2009$

A positive integer $n$ is given. On the segment $[0,n]$ of the real line $m$ distinct segments whose endpoints have integer coordinates are chosen. It turned out that it is impossible to choose some of thos segments such that their total length is $n$ and their union is $[0,n]$ Find the maximum possible value of $m$

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?

Define \[ N=\sum\limits_{k=1}^{60}e_k k^{k^k} \] where $e_k \in \{-1, 1\}$ for each $k$. Prove that $N$ cannot be the fifth power of an integer.

Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$

Let $n\geqslant 2$ be a fixed integer. Consider $n$ real numbers $a_1,a_2,\ldots,a_n$ not all equal and let\[d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.\]Determine in terms of $n{}$ the smalest and largest values the quotient $s/d$ may achieve. [i]Selected from the Kvant Magazine[/i]

Let $\mathbb{Z}_{\geq 0}$ be the set of nonnegative integers. Let $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$ be a function such that, for all $a,b \in \mathbb{Z}_{\geq0}$: \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\] Furthermore, suppose there exists $n \in \mathbb{Z}_{\geq0}$ such that $f(n)=577$. Let $S$ be the sum of all possible values of $f(2017)$. Find the remainder when $S$ is divided by $2017$. [i]Proposed by Zack Chroman[/i]

Ada and Emily are playing a game that ends when either player wins, after some number of rounds. Each round, either nobody wins, Ada wins, or Emily wins. The probability that neither player wins each round is $\frac{1}{5}$ and the probability that Emily wins the game as a whole is $\frac{3}{4}.$ If the probability that in a given round Emily wins is $\frac{m}{n}$ such that $m$ and $n$ are relatively prime integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$.

Find the number of ordered palindromic partitions of an integer $n$.

Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC \equal{} BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$ \textbf{(A)} \ \frac {1}{6} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {2}{3}$

Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression. Show that \[A_1 A_2 \ge G_1 G_2.\]

Following the previous set up, find the speed $v$ of the small block after it leaves the slope. (A) $v=v_\text{0}$ (B) $v=\frac{m}{m+M}v_\text{0}$ (C) $v=\frac{M}{m+M}v_\text{0}$ (D) $v=\frac{M-m}{m}v_\text{0}$ (E) $v=\frac{M-m}{m+M}v_\text{0}$

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.