Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying: (i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $ (ii) $\ N - a_i$ is a perfect square for $\ i = 1,2,3,4,5,6 $.

There are two urns each containing an arbitrary number of balls. We are allowed two types of operations: (a) remove an equal number of balls simultaneously from both the urns, (b) double the number of balls in any of them Show that after performing these operations finitely many times, both the urns can be made empty.

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find: a) The minimum value of $n$. b) For that value, find the least possible number of red squares.

Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$. (Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)

At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point. The losing team gets $0$ points. Draws do not occur. In the nal ranking, only one team turns out to have the least number of points (so there is no shared last place). Moreover, each team, except for the team having the least number of points, lost exactly one game against a team that got less points in the final ranking. a) Prove that the number of teams cannot be equal to $6$. b) Show, by providing an example, that the number of teams could be equal to $7$.

The arithmetic mean (average) of the first $n$ positive integers is: $ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $

For any point $ P$ on a segment $ AB$, isosceles and right-angled triangles $ AQP$ and $ PRB$ are constructed on the same side of $ AB$, with $ AP$ and $ PB$ as the bases. Determine the locus of the midpoint $ M$ of $ QR$ when $ P$ describes the segment $ AB$.

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccccc} & & & a & b & c & d \\ x & & & & & a & b \\ \hline & & c & d & b & d & b \\ + & c & e & b & f & b & \\ \hline & c & g & a & e & g & b \\ \end{tabular}$ [b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$? [b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$ [b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$, $b$, $c$, and $d$ be positive real numbers such that $a+b=c+d$ and $a^2+b^2>c^2+d^2$. Prove that $a^3+b^3>c^3+d^3$.

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Prove that there exist nonconstant polynomials $f, g$ with integer coefficients, such that for infinitely many primes $p$, $p \nmid f(x)-g(y)$ for any integers $x, y$.

In a language$,$ an alphabet with $25$ letters is used$;$ words are exactly all sequences of $($ not necessarily different $)$ letters of length $17.$ Two ends of a paper strip are glued so that the strip forms a ring$;$ the strip bears a sequence of $5^{18}$ letters$.$ Say that a word is singular if one can cut a piece bearing exactly that word from the strip$,$ but one cannot cut out two such non-overlapping pieces$.$ It is known that one can cut out $5^{16}$ non-overlapping pieces each containing the same word$.$ Determine the largest possible number of singular words$.$ [i](Bogdanov I.)[/i]

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

Find all non-negative integers $n$, $a$, and $b$ satisfying \[2^a + 5^b + 1 = n!.\]

Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties: $ \text{(i) } 2\lfloor x \rfloor \le f(x) \le 2 \lfloor x \rfloor +2,\quad\forall x\in (0,\infty ) $ $ \text{(ii) } f\circ f $ is monotone Can $ f $ be non-monotone? Justify.

Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 = \frac{1}{4}.$$ Prove that $$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$ [i]Proposed by Petar Filipovski, Macedonia[/i]

Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$? $\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old? (W. Janous, WRG Ursulinen, Innsbruck)

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

[b]p1.[/b] (See the diagram below.) $ABCD$ is a square. Points $G$, $H$, $I$, and $J$ are chosen in the interior of $ABCD$ so that: (i) $H$ is on $\overline{AG}$, $I$ is on $\overline{BH}$, $J$ is on $\overline{CI}$, and $G$ is on $\overline{DJ}$ (ii) $\vartriangle ABH \sim \vartriangle BCI \sim \vartriangle CDJ \sim \vartriangle DAG$ and (iii) the radii of the inscribed circles of $\vartriangle ABH$, $\vartriangle BCI$, $\vartriangle CDJ$, $\vartriangle DAK$, and $GHIJ$ are all the same. What is the ratio of $\overline{AB}$ to $\overline{GH}$? [img]https://cdn.artofproblemsolving.com/attachments/f/b/47e8b9c1288874bc48462605ecd06ddf0f251d.png[/img] [b]p2.[/b] The three solutions $r_1$, $r_2$, and $r_3$ of the equation $$x^3 + x^2 - 2x - 1 = 0$$ can be written in the form $2 \cos (k_1 \pi)$, $2 \cos (k_2 \pi)$, and $2 \cos (k_3 \pi)$ where $0 \le k_1 < k_2 < k_3 \le 1$. What is the ordered triple $(k_1, k_2, k_3)$? [b]p3.[/b] $P$ is a convex polyhedron, all of whose faces are either triangles or decagons ($10$-sided polygon), though not necessarily regular. Furthermore, at each vertex of $P$ exactly three faces meet. If $P$ has $20$ triangular faces, how many decagonal faces does P have? [b]p4.[/b] $P_1$ is a parabola whose line of symmetry is parallel to the $x$-axis, has $(0, 1)$ as its vertex, and passes through $(2, 2)$. $P_2$ is a parabola whose line of symmetry is parallel to the $y$-axis, has $(1, 0)$ as its vertex, and passes through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for all odd $ n > 1$, we have $ 8n \plus{} 4|C^{4n}_{2n}$.