[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A_1$, $A_2$, $\ldots$, $A_n$, $n\geq 2$ be $n$ finite sets with the properties i) $|A_i| \geq 2$, for all $1\leq i \leq n$; ii) $|A_i\cap A_j| \neq 1$, for all $1\leq i<j\leq n$. Prove that the elements of the set $\displaystyle \bigcup_{i=1}^n A_i$ can be colored with 2 colors, such that all the sets $A_i$ are bi-color, for all $1\leq i \leq n$.

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

[b]p1.[/b] Isosceles trapezoid $ABCD$ has $AB = 2$, $BC = DA =\sqrt{17}$, and $CD = 4$. Point $E$ lies on $\overline{CD}$ such that $\overline{AE}$ splits $ABCD$ into two polygons of equal area. What is $DE$? [b]p2.[/b] At the Berkeley Sandwich Parlor, the famous BMT sandwich consists of up to five ingredients between the bread slices. These ingredients can be either bacon, mayo, or tomato, and ingredients of the same type are indistiguishable. If there must be at least one of each ingredient in the sandwich, and the order in which the ingredients are placed in the sandwich matters, how many possible ways are there to prepare a BMT sandwich? [b]p3.[/b] Three mutually externally tangent circles have radii $2$, $3$, and $3$. A fourth circle, distinct from the other three circles, is tangent to all three other circles. The sum of all possible radii of the fourth circle can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p_1,p_2, p_3$ be positive numbers such that $p_1 + p_2 + p_3 = 1$. If $a_1 <a_2 <a_3$ and $b_1 <b_2 <b_3$ prove that $$(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)$$

Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$

Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear. (Malik Talbi)

Let $p>q$ be primes, such that $240 \nmid p^4-q^4$. Find the maximal value of $\frac{q} {p}$.

Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.) (translated by Miklós Maróti)

Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]

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 quadrilateral and a pentagon (both not self-intersecting) intersect each other at $N$ distinct points, where $N$ is a positive integer. What is the maximal possible value of $N$? [i]Proposed by James Lin

What is the value of $(2^0-1+5^2+0)^{-1}\times 5$? $\textbf{(A) }-125\qquad\textbf{(B) }-120\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac5{24}\qquad\textbf{(E) }25$

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let $\{u_{n}\}_{n \ge 0}$ be a sequence of positive integers defined by \[u_{0}= 1, \;u_{n+1}= au_{n}+b,\] where $a, b \in \mathbb{N}$. Prove that for any choice of $a$ and $b$, the sequence $\{u_{n}\}_{n \ge 0}$ contains infinitely many composite numbers.

The permutations of $OLYMPIAD$ are arranged in lexicographical order, with $ADILMOPY$ being arrangement 1 and its reverse being arrangement $40320$. Yu Semo and Yu Sejmo both choose a uniformly random arrangement. The immature Yu Sejmo exclaims, ``My fourth letter is $L$!" while Yu Semo remains silent. Given this information, let $E_1$ be the expected arrangement number of Yu Semo and $E_2$ be the expected arrangement number of Yu Sejmo. Compute $E_2 - E_1$.

Two players take turns placing an unused number from {1, 2, 3, 4, 5, 6, 7, 8} into one of the empty squares in the array to the right. The game ends once all the squares are filled. The first player wins if the product of the numbers in the top row is greater. The second player wins if the product of the numbers in the bottom row is greater. If both players play with perfect strategy, who wins this game? [asy] unitsize(32); int[][] a = { {1, 2, 3, 4}, {5, 6, 7, 8}}; for (int i = 0; i < 4; ++i) { for (int j = 0; j < 2; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(16pt)); } } [/asy]

For any positive integer $n$, let $p(n)$ be the product of its digits in base-$10$ representation. Find the maximum possible value of $\frac{p(n)}{n}$ over all integers $n \ge 10$.

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.

Let $j$ and $k$ be integers. Prove that the inequality $$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.

Prove that there exist infinitely many pairwisely disjoint sets $A(1), A(2),...,A(2014)$ which are not empty, whose union is the set of positive integers and which satisfy the following condition: For arbitrary positive integers $a$ and $b$, at least two of the numbers $a$, $b$ and $GCD(a,b)$ belong to one of the sets $A(1), A(2),...,A(2014)$.

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

How many positive integers have exactly three proper divisors, each of which is less than 50?

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

Equilateral triangle $\vartriangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\vartriangle BFD$ and $\vartriangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$ and $b$ squarefree, find $a + b + c$.