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 $7$?

Find all the pairs $(a,b)$ of integer numbers such that: $\triangleright$ $a-b-1|a^2+b^2$ $\triangleright$ $\frac{a^2+b^2}{2ab-1}=\frac{20}{19}$

Prove that for any positive reals $ a,b,c,d $ with $ a+b+c+d = 4 $, we have $$ \sum\limits_{cyc}{\frac{3a^3}{a^2+ab+b^2}}+\sum\limits_{cyc}{\frac{2ab}{a+b}} \ge 8 $$

Let $C$ be the curve $y = x^3$ (where $x$ takes all real values). The tangent at $A$ meets the curve again at $B.$ Prove that the gradient at $B$ is $4$ times the gradient at $A.$

Suppose you are given six colours and, are asked to colour each face of a cube by a different colour. Determine the different number of colouring possible.

Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)

Find the sum of the greatest common factor and the least common multiple of $12$ and $18$.

Let $S(b)$ be the number of nonuples of positive integers $(a_1, a_2, \ldots , a_9)$ satisfying $3b-1=a_1+a_2+\ldots+a_9$ and $b^2+1=a_1^2+\ldots+a_9^2$. Prove that for all $\epsilon>0$, there exists $C_{\epsilon}>0$ such that $S(b)\leq C_{\epsilon}b^{3+\epsilon}$.

Find all the real numbers $ \alpha$ satisfy the following property: for any positive integer $ n$ there exists an integer $ m$ such that $ \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}$.

Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? $\textbf{(A)}~2\sqrt{11}\qquad\textbf{(B)}~4\sqrt{3}\qquad\textbf{(C)}~8\qquad\textbf{(D)}~4\sqrt{5}\qquad\textbf{(E)}~9$

Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.

Positive integers $k, n$ and subsets $A_1, A_2, ..., A_k$ of the set $\{1, 2, ..., 2n\}$ are given. We will say that a pair of numbers $x, y$ is good, if $x<y$, $x, y\in \{1, 2, ..., 2n\}$ and there exists exactly one index $i\in \{1, 2, ..., 2n\}$, for which exactly one of $x, y$ belongs to $A_i$. Prove that there are at most $n^2$ good pairs.

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: [i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order; [i](ii)[/i] the polygon is circumscribable about a circle. [i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.

A circle $k = (S, r)$ is given and a hexagon $AA'BB'CC'$ inscribed in it. The lengths of sides of the hexagon satisfy $AA' = A'B, BB' = B'C, CC' = C'A$. Prove that the area $P$ of triangle $ABC$ is not greater than the area $P'$ of triangle $A'B'C'$. When does $P = P'$ hold?

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Let $a, b, c$ be the sides of a triangle such that $\frac{a^2+b^2+c^2}{ ab+bc+ca}$ is an integer. Find the relation between $a, b, c$.

Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$. $Turkey$

Let $a,b,c$ be three integers for which the sum \[ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\] is integer. Prove that each of the three numbers \[ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}\] is integer. (Proposed by Gerhard J. Woeginger)

(a) Prove that for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$, \[ \max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0. \] (b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$.

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a [i]dime for a dozen[/i] pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend [b]all[/b] $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I [i]give my two cents[/i] to Bill, he then notes that he can instead spend [b]all[/b] $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible? Note: Both times Bill must spend [b]all[/b] of his cents on pencil packages, i.e. have zero cents after either purchase.