Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.

Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \] [i]Proposed by Andria Gvaramia, Georgia [/i]

A quadratic polynomial with positive integer coefficients and rational roots can be written as $196x^2+Bx + 135$ for some integer $B.$ What is the sum of all possible values of $B$ such that $\gcd(B, 196 \cdot 135) = 1$?

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer coefficient polynomial $P(x)$ such that for all positive integer $x$, we have $$\tau(P(x))\geq\tau(x)$$Where $\tau(n)$ denotes the number of divisors of $n$. Define $\tau(0)=\infty$. Note: you can use this conclusion. For all $\epsilon\geq0$, there exists a positive constant $C_\epsilon$ such that for all positive integer $n$, the $n$th smallest prime is at most $C_\epsilon n^{1+\epsilon}$. [i]Proposed by USJL[/i]

A three-digit number has, from left to right, the digits $ h$, $ t$, and $ u$, with $ h>u$. When the number with the digits reversed is subtracted from the original number, the units' digit in the difference of $ r$. The next two digits, from right to left, are: $ \textbf{(A)}\ \text{5 and 9} \qquad \textbf{(B)}\ \text{9 and 5} \qquad \textbf{(C)}\ \text{impossible to tell} \qquad \textbf{(D)}\ \text{5 and 4} \qquad \textbf{(E)}\ \text{4 and 5}$

[b]p1.[/b] What is the last digit of $$1! + 2! + ... + 10!$$ where $n!$ is defined to equal $1 \cdot 2 \cdot ... \cdot n$? [b]p2.[/b] What pair of positive real numbers, $(x, y)$, satisfies $$x^2y^2 = 144$$ $$(x - y)^3 = 64?$$ [b]p3.[/b] Paul rolls a standard $6$-sided die, and records the results. What is the probability that he rolls a $1$ ten times before he rolls a $6$ twice? [b]p4.[/b] A train is approaching a $1$ kilometer long tunnel at a constant $40$ km/hr. It so happens that if Roger, who is inside, runs towards either end of the tunnel at a contant $10$ km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger? [b]p5.[/b] Let $ABC$ be a triangle with $A$ being a right angle. Let $w$ be a circle tangent to $\overline{AB}$ at $A$ and tangent to $\overline{BC}$ at some point $D$. Suppose $w$ intersects $\overline{AC}$ again at $E$ and that $\overline{CE} = 3$, $\overline{CD} = 6$. Compute $\overline{BD}$. [b]p6.[/b] In how many ways can $1000$ be written as a sum of consecutive integers? [b]p7.[/b] Let $ABC$ be an isosceles triangle with $\overline{AB} = \overline{AC} = 10$ and $\overline{BC} = 6$. Let $M$ be the midpoint of $\overline{AB}$, and let $\ell$ be the line through $A$ parallel to $\overline{BC}$. If $\ell$ intersects the circle through $A$, $C$ and $M$ at $D$, then what is the length of $\overline{AD}$? [b]p8.[/b] How many ordered triples of pairwise relatively prime, positive integers, $\{a, b, c\}$, have the property that $a + b$ is a multiple of $c$, $b + c$ is a multiple of $a$, and $a + c$ is a multiple of $b$? [b]p9.[/b] Consider a hexagon inscribed in a circle of radius $r$. If the hexagon has two sides of length $2$, two sides of length $7$, and two sides of length $11$, what is $r$? [b]p10.[/b] Evaluate $$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0} \frac{\left( (-1)^i + (-1)^j\right) \cos (i) \sin (j)}{i!j!} ,$$ where angles are measured in degrees, and $0!$ is defined to equal $1$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

Point $D$ is inside $\triangle ABC$ and $AD=DC$. $BD$ intersect $AC$ in $E$. $\frac{BD}{BE}=\frac{AE}{EC}$. Prove, that $BE=BC$

Let $n$ be a positive integer and let $a_n$ be the number of ways to write $n$ as a sum of positive integers, such that any two summands differ by at least $2$. Also, let $b_n$ be the number of ways to write $n$ as a sum of positive integers of the form $5k\pm 1$, $k \in Z$. Prove that $\frac{a_n}{b_n}$ is a constant for all positive integers $n$.

In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.

Find the smallest positive integer $N$ satisfying the following three properties. $\bullet$ N leaves a remainder of $5$ when divided by $7$. $\bullet$ N leaves a remainder of $6$ when divided by $ 8$. $\bullet$ N leaves a remainder of $7$ when divided by $9$.

Let $S$ be a set of size $11$. A random $12$-tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

In a right circular cone, $A$ is the vertex, $B$ is the center of the base, and $C$ is a point on the circumference of the base with $BC = 1$ and $AB = 4$. There is a trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$. A right circular cylinder whose surface contains the points $A, C$, and $D$ intersects the cone such that its axis of symmetry is perpendicular to the plane of the trapezoid, and $\overline{CD}$ is a diameter of the cylinder. A sphere radius $r$ lies inside the cone and inside the cylinder. The greatest possible value of $r$ is $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c + d$.

A sequence of integers $a_{1}, a_{2}, \cdots$ is called $good$ if: • $a_{1}=1$, and; • $a_{i+1}-a_{i}$ is either $1$ or $2$ for all $i \geq 1$. Find all positive integers n that cannot be written as a sum $n = a_{1} + a_{2} + \cdots + a_{k}$, such that the integers $a_{1} , a_{2} , \cdots , a_{k}$ forms a good sequence.

Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.

There are $n$ different integers on the blackboard. Whenever two of these integers are chosen, either their sum or difference (possibly both) will be a positive integral power of $2$. Find the greatest possible value of $n$.