Find the positive integer $n$ such that $$1 + 2 + 3 +...+ n = (n + 1) + (n + 2) +...+ (n + 35).$$

Prove that for any positive integer $n$, \[\left\lfloor \frac{n+1}{2}\right\rfloor+\left\lfloor \frac{n+2}{4}\right\rfloor+\left\lfloor \frac{n+4}{8}\right\rfloor+\left\lfloor \frac{n+8}{16}\right\rfloor+\cdots = n.\]

Study the real function $f(x) = \left(1 +\frac{1}{x}\right)^x$ defined for $ x \in R - \{-1, 0\}$ . Graphic representation.

The average age of $33$ fifth-graders is $11$. The average age of $55$ of their parents is $33$. What is the average age of all of these parents and fifth-graders? $\textbf{(A) }22\qquad\textbf{(B) }23.25\qquad\textbf{(C) }24.75\qquad\textbf{(D) }26.25\qquad\textbf{(E) }28$

Find the number of ordered pairs $(x,y)$ of integers with $0 \le x < 2023$ and $0 \le y < 2023$ such that $y^3 \equiv x^2 \pmod{2023}.$

The function $f$ is defined on positive integers : if $n$ has prime factorization $p^{k_1}_{1} p^{k_2}_{2} ...p^{k_t}_{t}$ then $f(n) = (p_1-1)^{k_1+1}(p_2-1)^{k_2+1}...(p_t-1)^{k_t+1}$. If we keep using this function repeatedly, starting from any positive integer $n$, we will always get to $1$ after some number of steps. What is the smallest integer $n$ for which we need exactly $6$ steps to get to $1$?

The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button ``+": [img]6965[/img] Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button ``+". $A$ passes the calculator to $B$, which presses a digit in the same row or column with the one pressed by $A$ that is not the same as the last one pressed by $A$; and then presses + and returns the calculator to $A$, repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?

A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap. How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions: - each letter may appear no more than twice in each palindrome, - the length of each palindrome is at least $3$ letters. (Any possible combination of letters is considered a word.)

(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$.

Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4. What is the magnitude of the $z$-coordinate of $T_0$?

Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.

* Assume that the number of a tree’s leaves is a multiple of $15$. Neglecting the shade of the trunk and branches prove that one can rip off the tree $7/15$ of its leaves so that not less than $8/15$ of its shade remains.

Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

Let be a set $ A\in (0,\infty )\setminus\{ 1\} $ and two operations $ *,\circ :A^2\longrightarrow A $ defined as $$ x*y=x^{2\log_3 y} ,\quad x\circ y= x^{3\log_2y} , $$ and chosen such that $ (A,*) , (A,\circ ) $ are groups. Prove that these groups are isomorphic. [i]Gabriel Iorgulescu[/i]

How many solutions of the equation $\tan x = \tan \tan x$ are on the interval $0 \le x \le \tan^{-1} 942$? (Here $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.)

Prove that for positive real numbers $a, b$ and $c$ the inequality $2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2$ holds if and only if $a,b,c$ are the sides of a triangle.

Let $a\geq 2$ a fixed natural number, and let $a_{n}$ be the sequence $a_{n}=a^{a^{.^{.^{a}}}}$ (e.g $a_{1}=a$, $a_{2}=a^a$, etc.). Prove that $(a_{n+1}-a_{n})|(a_{n+2}-a_{n+1})$ for every natural number $n$.

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$

Let $m$ be a positive integer, and real numbers $a_1, a_2,\ldots , a_m$ satisfy \[\frac{1}{m}\sum_{i=1}^{m}a_i = 1,\] \[\frac{1}{m}\sum_{i=1}^{m}a_i ^2= 11,\] \[\frac{1}{m}\sum_{i=1}^{m}a_i ^3= 1,\] \[\frac{1}{m}\sum_{i=1}^{m}a_i ^4= 131.\] Prove that $m$ is a multiple of $7$. [i] Proposed by usjl[/i]

[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you? [b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle. [b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties: • $0 \in S$ • If $x \in S$ then $n - 2 - x \in S$ • If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$. Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime. (Note: “$\in$” means “belongs to”.) [b]p4.[/b] Consider the $3 \times 3$ grid shown below $\begin{tabular}{|l|l|l|l|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{tabular}$ A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move. Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.) [b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?

Dene the function $f : R \to R$ by $$f(x) =\begin{cases} \dfrac{1}{x^2+\sqrt{x^4+2x}}\,\,\, \text{if} \,\,\,x \notin (- \sqrt[3]{2}, 0] \\ \,\,\, 0 \,\,\,, \,\,\, \text{otherwise} \end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x) = 1$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b,c, d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d.$ (Here, $f^n(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^3(x) = f(f(f(x)))$.)

(B.Frenkin, 8) Does a regular polygon exist such that just half of its diagonals are parallel to its sides?

Let $a,b,c,d,m,n$ be positive real numbers. $P=\sqrt{ab}+\sqrt{cd},Q=\sqrt{ma+nc}\cdot\sqrt{\frac{b}{m}+\frac{d}{n}}$. Then $\text{(A)}P\geq Q\qquad\text{(B)}P\leq Q\qquad\text{(C)}P<Q\qquad\text{(D)}$Not sure

During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.