What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$

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