Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

Positive real numbers $a,b,c$ satisfy $$2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2.$$ Prove that $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \geq(ab+bc+ca)^2.$$

$\triangle ABC$ has $AB=4$ and $AC=6$. Let point $D$ be on line $AB$ so that $A$ is between $B$ and $D$. Let the angle bisector of $\angle BAC$ intersect line $BC$ at $E$, and let the angle bisector of $\angle DAC$ intersect line $BC$ at $F$. Given that $AE=AF$, find the square of the circumcircle's radius' length.

For each pair $(\alpha, \beta)$ of non-negative reals with $\alpha+\beta \geq 2$, determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(x)f(y) \leq f(xy)+\alpha x+\beta y$$ for all reals $x, y$.

Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.

Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number). (a) What is the smallest and biggest representable number? (b) Construct 45 representable numbers.

At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? $\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64$

On a $2004\times 2004$ chessboard we place $2004$ white knights$^1$ in the upper row, and $2004$ black ones in the lowest row. After a finite number of regular chess moves$^2$ , we get the opposite situation where the black ones are on the top and the white ones on the bottom lines. In a [i]turn [/i] we make a move with each of the pieces of a color. If you know that each square except those on which the knights originally lie, must not be used more than once in this process, and that after each turn no $2$ knights of the same color can be attacking each other$^3$ , determine the number of ways in which this can be accomplished. $^1$ also known as horses $^2$ the knight can be moved either one square horizontally and two vertically or two squares horizontally and one vertically, in any direction on both horizontal and vertical lines $^3$ a knight is attacking another knight, if in one chess move, the first one can be placed on the second one’s place

In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.

[u]Set 1[/u] [b]p1.[/b] Arnold is currently stationed at $(0, 0)$. He wants to buy some milk at $(3, 0)$, and also some cookies at $(0, 4)$, and then return back home at $(0, 0)$. If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take? [b]p2.[/b] Dilhan selects $1$ shirt out of $3$ choices, $1$ pair of pants out of $4$ choices, and $2$ socks out of $6$ differently-colored socks. How many outfits can Dilhan select? All socks can be worn on both feet, and outfits where the only difference is that the left sock and right sock are switched are considered the same. [b]p3.[/b] What is the sum of the first $100$ odd positive integers? [b]p4.[/b] Find the sum of all the distinct prime factors of $1591$. [b]p5.[/b] Let set $S = \{1, 2, 3, 4, 5, 6\}$. From $S$, four numbers are selected, with replacement. These numbers are assembled to create a $4$-digit number. How many such $4$-digit numbers are multiples of $3$? [u]Set 2[/u] [b]p6.[/b] What is the area of a triangle with vertices at $(0, 0)$, $(7, 2)$, and $(4, 4)$? [b]p7.[/b] Call a number $n$ “warm” if $n - 1$, $n$, and $n + 1$ are all composite. Call a number $m$ “fuzzy” if $m$ may be expressed as the sum of $3$ consecutive positive integers. How many numbers less than or equal to $30$ are warm and fuzzy? [b]p8.[/b] Consider a square and hexagon of equal area. What is the square of the ratio of the side length of the hexagon to the side length of the square? [b]p9.[/b] If $x^2 + y^2 = 361$, $xy = -40$, and $x - y$ is positive, what is $x - y$? [b]p10.[/b] Each face of a cube is to be painted red, orange, yellow, green, blue, or violet, and each color must be used exactly once. Assuming rotations are indistinguishable, how many ways are there to paint the cube? [u]Set 3[/u] [b]p11.[/b] Let $D$ be the midpoint of side $BC$ of triangle $ABC$. Let $P$ be any point on segment $AD$. If $M$ is the maximum possible value of $\frac{[PAB]}{[PAC]}$ and $m$ is the minimum possible value, what is $M - m$? Note: $[PQR]$ denotes the area of triangle $PQR$. [b]p12.[/b] If the product of the positive divisors of the positive integer $n$ is $n^6$, find the sum of the $3$ smallest possible values of $n$. [b]p13.[/b] Find the product of the magnitudes of the complex roots of the equation $(x - 4)^4 +(x - 2)^4 + 14 = 0$. [b]p14.[/b] If $xy - 20x - 16y = 2016$ and $x$ and $y$ are both positive integers, what is the least possible value of $\max (x, y)$? [b]p15.[/b] A peasant is trying to escape from Chyornarus, ruled by the Tsar and his mystical faith healer. The peasant starts at $(0, 0)$ on a $6 \times 6$ unit grid, the Tsar’s palace is at $(3, 3)$, the healer is at $(2, 1)$, and the escape is at $(6, 6)$. If the peasant crosses the Tsar’s palace or the mystical faith healer, he is executed and fails to escape. The peasant’s path can only consist of moves upward and rightward along the gridlines. How many valid paths allow the peasant to escape? PS. You should use hide for answers. Rest sets have been posted [url=https://artofproblemsolving.com/community/c3h2784259p24464954]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.

A sequence of integers $a_0,\ a_1,\dots a_n \dots $ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$, and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$

In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

What are the last two digits of $$2^{3^{4^{...^{2019}}}} ?$$

A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

Find all triples of positive integers $(x,y,z)$ that satisfy the equation $$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$

Let $ABC$ be an equilateral triangle with circumcircle of center $O$ and radius $R$. Point $M$ is exterior to the triangle such that $S_bS_c = S_aS_b+S_aS_c$, where $S_a, S_b, S_c$ are the areas of triangles $MBC,MCA,MAB$ respectively. Prove that $OM \ge R$. Nguyễn Tiến Lâm

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Given the rectangle $ABCD$. The points $E ,F$ lie on the segments $(BC) , (DC)$ respectively, such that $\angle DAF = \angle FAE$. Proce that if $DF + BE = AE$ then $ABCD$ is square.

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

What is the product of the factors of $30^{12}$ that are congruent to $1$ modulo $7$?