The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

$\definecolor{A}{RGB}{70,80,0}\color{A}\fbox{C4.}$ Show that for any positive integer $n \ge 3$ and some subset of $\lbrace{1, 2, . . . , n}\rbrace$ with size more than $\frac{n}2 + 1$, there exist three distinct elements $a, b, c$ in the subset such that $$\definecolor{A}{RGB}{255,70,255}\color{A} (ab)^2 + (bc)^2 + (ca)^2$$is a perfect square. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1736[/color]

The exchange rates of the Dollar and the German mark during the week changed as follows: $\begin{tabular}{|l|l|l|} \hline & Dollar & Mark \\ \hline Monday & 4000 rub. & 2500 rub. \\ \hline Tuesday & 4500 rub. & 2800 rub.\\ \hline Wednesday & 5000 rub. & 2500 rub.\\ \hline Thursday & 4500 rub. & 3000 rub.\\ \hline Friday & 4000 rub. & 2500 rub.\\ \hline Saturday & 4500 rub. & 3000 rub.\\ \hline \end{tabular}$ What percentage was the maximum possible increase in capital this week by playing on changes in the exchange rates of these currencies? (The initial capital was in rubles. The final capital should also be in rubles. During the week, the available money can be distributed as desired into rubles, dollars and marks. The selling and purchasing rates are considered the same.)

Decomposition of number $n$ is showing $n$ as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer $n$ show that number of decompositions is $2^{n-1}$

[b]p1.[/b] Find the number of zeroes at the end of $20^{17}$. [b]p2.[/b] Express $\frac{1}{\sqrt{20} +\sqrt{17}}$ in simplest radical form. [b]p3.[/b] John draws a square $ABCD$. On side $AB$ he draws point $P$ so that $\frac{BP}{PA}=\frac{1}{20}$ and on side $BC$ he draws point $Q$ such that $\frac{BQ}{QC}=\frac{1}{17}$ . What is the ratio of the area of $\vartriangle PBQ$ to the area of $ABCD$? [b]p4.[/b] Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints? [b]p5.[/b] Alex is playing a game with an unfair coin which has a $\frac15$ chance of flipping heads and a $\frac45$ chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins? [b]p6.[/b] Positive two-digit number $\overline{ab}$ has $8$ divisors. Find the number of divisors of the four-digit number $\overline{abab}$. [b]p7.[/b] Call a positive integer $n$ diagonal if the number of diagonals of a convex $n$-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to $2017$. [b]p8.[/b] There are $4$ houses on a street, with $2$ on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors. [b]p9.[/b] Compute $$|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.$$ [b]p10.[/b] Given points $A,B$ in the coordinate plane, let $A \oplus B$ be the unique point $C$ such that $\overline{AC}$ is parallel to the $x$-axis and $\overline{BC}$ is parallel to the $y$-axis. Find the point $(x, y)$ such that $((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y)$. [b]p11.[/b] In the following subtraction problem, different letters represent different nonzero digits. $\begin{tabular}{ccccc} & M & A & T & H \\ - & & H & A & M \\ \hline & & L & M & T \\ \end{tabular}$ How many ways can the letters be assigned values to satisfy the subtraction problem? [b]p12.[/b] If $m$ and $n$ are integers such that $17n +20m = 2017$, then what is the minimum possible value of $|m-n|$? [b]p13. [/b]Let $f(x)=x^4-3x^3+2x^2+7x-9$. For some complex numbers $a,b,c,d$, it is true that $f (x) = (x^2+ax+b)(x^2+cx +d)$ for all complex numbers $x$. Find $\frac{a}{b}+ \frac{c}{d}$. [b]p14.[/b] A positive integer is called an imposter if it can be expressed in the form $2^a +2^b$ where $a,b$ are non-negative integers and $a \ne b$. How many almost positive integers less than $2017$ are imposters? [b]p15.[/b] Evaluate the infinite sum $$\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...$$ [b]p16.[/b] Each face of a regular tetrahedron is colored either red, green, or blue, each with probability $\frac13$ . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors? [b]p17.[/b] Let $(k,\sqrt{k})$ be the point on the graph of $y=\sqrt{x}$ that is closest to the point $(2017,0)$. Find $k$. [b]p18.[/b] Alice is going to place $2016$ rooks on a $2016 \times 2016$ chessboard where both the rows and columns are labelled $1$ to $2016$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations. [b]p19.[/b] Let $f (n)$ be a function defined recursively across the natural numbers such that $f (1) = 1$ and $f (n) = n^{f (n-1)}$. Find the sum of all positive divisors less than or equal to $15$ of the number $f (7)-1$. [b]p20.[/b] Find the number of ordered pairs of positive integers $(m,n)$ that satisfy $$gcd \,(m,n)+ lcm \,(m,n) = 2017.$$ [b]p21.[/b] Let $\vartriangle ABC$ be a triangle. Let $M$ be the midpoint of $AB$ and let $P$ be the projection of $A$ onto $BC$. If $AB = 20$, and $BC = MC = 17$, compute $BP$. [b]p22.[/b] For positive integers $n$, define the odd parent function, denoted $op(n)$, to be the greatest positive odd divisor of $n$. For example, $op(4) = 1$, $op(5) = 5$, and $op(6) =3$. Find $\sum^{256}_{i=1}op(i).$ [b]p23.[/b] Suppose $\vartriangle ABC$ has sidelengths $AB = 20$ and $AC = 17$. Let $X$ be a point inside $\vartriangle ABC$ such that $BX \perp CX$ and $AX \perp BC$. If $|BX^4 -CX^4|= 2017$, the compute the length of side $BC$. [b]p24.[/b] How many ways can some squares be colored black in a $6 \times 6$ grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct. [b]p25.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = 2$, $AD = 4$, and $\angle ABC = 120^o$. Let $M$ be the midpoint of $BD$. If $\angle AMC = 90^o$, find the length of segment $CD$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n \plus{} 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n \plus{} p \plus{} q, n)$ are all nice.

$x^2 + 2y^2 = 1$ solve in integers.

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

Morteza wishes to take two real numbers $S$ and $P$, and then to arrange six pairwise distinct real numbers on a circle so that for each three consecutive numbers at least one of the two following conditions holds: 1) their sum equals $S$ 2) their product equals $P$. Determine if Morteza’s wish could be fulfilled.

Suppose that $A$ is a set of positive integers less than $N$ and that no two distinct elements of $A$ sum to a perfect square. That is, if $a_1, a_2 \in A$ and $a_1 \neq a_2$ then $|a_1+a_2|$ is not a square of an integer. Prove that the maximum number of elements in $A$ is at least $\left\lfloor\frac{11}{32}N\right\rfloor$ .

A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

In $\vartriangle ABC$, the external angle bisector of $\angle BAC$ intersects line $BC$ at $D$. $E$ is a point on ray $\overrightarrow{AC}$ such that $\angle BDE = 2\angle ADB$. If $AB = 10$, $AC = 12$, and $CE = 33$, compute $\frac{DB}{DE}$ .

Let $I$ be the incenter of triangle $ABC$ and let $D$ be the midpoint of side $AB$. Prove that if the angle $\angle AOD$ is right, then $AB+BC=3AC$. [I]Proposed by S. Ivanov[/i]

Between any two cities of a country there is only one one-way road. Show that there is a city from that every other city can be reached directly or by going over only one intermediate city. [hide] I'm sure it was posted before but couldn't find it. [/hide]

For non negative reals $a,b$ we know that $a^2+a+b^2\ge a^4+a^3+b^4$. Prove that $$\frac{1-a^4}{a^2}\ge \frac{b^2-1}{b}$$

The circumradius of acute triangle $ABC$ is twice of the distance of its circumcenter to $AB$. If $|AC|=2$ and $|BC|=3$, what is the altitude passing through $C$? $ \textbf{(A)}\ \sqrt {14} \qquad\textbf{(B)}\ \dfrac{3}{7}\sqrt{21} \qquad\textbf{(C)}\ \dfrac{4}{7}\sqrt{21} \qquad\textbf{(D)}\ \dfrac{1}{2}\sqrt{21} \qquad\textbf{(E)}\ \dfrac{2}{3}\sqrt{14} $

Each of the $29$ people attending a party wears one of three different types of hats. Call a person [i]lucky[/i] if at least two of his friends wear different types of hats. Show that it is always possible to replace the hat of a person at this party with a hat of one of the other two types, in a way that the total number of lucky people is not reduced.

Consider all trapezoids in a coordinate plane with interior angles of $90^o, 90^o, 45^o$ and $135^o$ whose bases are parallel to a coordinate axis and whose vertices have integer coordinates. Define the [i]size [/i] of such a trapezoid as the total number of points with integer coordinates inside and on the boundary of the trapezoid. (a) How many pairwise non-congruent such trapezoids of size $2001$ are there? (b) Find all positive integers not greater than $50$ that do not appear as sizes of any such trapezoid.

Find all triples of reals $(x,y,z)$ satisfying: $$\begin{cases} \frac{1}{3} \min \{x,y\} + \frac{2}{3} \max \{x,y\} = 2017 \\ \frac{1}{3} \min \{y,z\} + \frac{2}{3} \max \{y,z\} = 2018 \\ \frac{1}{3} \min \{z,x\} + \frac{2}{3} \max \{z,x\} = 2019 \\ \end{cases}$$

For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$ Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that [list=disc] [*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and [*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$. [/list] Prove that $\max(a_1,a_{2023})\ge 507$.

Let $m,n$ be distinct positive integers. Prove that $$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$ Further, determine when equality holds.