[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ + & & & d & e \\ \hline & f & a & g & c \\ x & b & b & h & \\ \hline f & f & e & g & c \\ \end{tabular}$ [b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m? [b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse? [img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

Consider an infinite sequence consisting of distinct positive integers such that each term (except the rst one) is either an arithmetic mean or a geometric mean of two neighboring terms. Does it necessarily imply that starting at some point the sequence becomes either arithmetic progression or a geometric progression?

Let $P:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $P(X)=X$ has no real solution. Prove that $P(P(X))=X$ has no real solution.

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

Let $p(x)$ and $q(x)$ non constant real polynomials of degree at most $n$ ($n > 1$). Show that there exists a non zero polynomial $F(x,y)$ in two variables with real coefficients of degree at most $2n-2,$ such that $F(p(t),q(t)) = 0$ for every $t\in \mathbb{R}$.

Find all pairs $(a,b,c)$ of positive integers such that $$\frac{a}{2^a}=\frac{b}{2^b}+\frac{c}{2^c}$$

Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions: (i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$, (ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$, (iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.

Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$.

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1 \le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1 \le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which \[N(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)})\le \min(N(a_1,a_2,\ldots,a_n),N(b_1,b_2,\ldots,b_n)).\]

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $\mathbb R^+\to\mathbb R^+$ such that \[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d), \quad \forall a,b,c,d\in\mathbb R^+; \quad abcd=1\]

Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$. The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$. Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.

Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Powell Zhang[/i]

Find the necessary and sufficient conditions that the real numbers $ a $, $ b $, $ c $ should satisfy so that the equation $$x^3 + ax^2 + bx + c = 0$$ has three real roots creating an arithmetic progression.

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.

Which of the two numbers is bigger : $\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?

If $x +\frac{1}{x} = 5$, what is the value of $x^3 +\frac{1}{x^3} $ ?

Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$

The quadratic equation $ x^2 \plus{} mx \plus{} n \equal{} 0$ has roots that are twice those of $ x^2 \plus{} px \plus{} m \equal{} 0$, and none of $ m,n,$ and $ p$ is zero. What is the value of $ n/p$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$. Calculate $[M(a,b)]$ (ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.