Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$ , sum of some $a_i$-s is $j$. [i]Proposed by Navid Safaei [/i]

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

At an international event there are $100$ countries participating, each with its own flag. There are $10$ distinct flagpoles at the stadium, labelled 1,#2,...,#10 in a row. In how many ways can all the $100$ flags be hoisted on these $10$ flagpoles, such that for each $i$ from $1$ to $10$, the flagpole #i has at least $i$ flags? (Note that the vertical order of the flagpoles on each flag is important)

A set of five different positive integers is called [i]virtual[/i] if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors. [i]Proposed by Víctor Almendra[/i]

Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$.

In how many ways can Alice, Bob, Charlie, David, and Eve split $18$ marbles among themselves so that no two of them have the same number of marbles?

Prove that $16<\sum_{i=1}^{80}\frac{1}{\sqrt{i}}<17$.

Suppose that $A$ is a set of real numbers between $3$ and $2024$ inclusive such that for any $x, y \in A$ with $x \neq y,$ we have $|x-y|>\tfrac{xy}{2+2xy}.$ What is the largest possible size of $A$?

p1. Find all real numbers $x$ that satisfy the inequality $$\frac{x^2-3}{x^2-1}+ \frac{x^2 + 5}{x^2 + 3} \ge \frac{x^2-5}{x^2-3}+\frac{x^2 + 3}{x^2 + 1}$$ p2. It is known that $m$ is a four-digit natural number with the same units and thousands digits. If $m$ is a square of an integer, find all possible numbers $m$. p3. In the following figure, $\vartriangle ABP$ is an isosceles triangle, with $AB = BP$ and point $C$ on $BP$. Calculate the volume of the object obtained by rotating $ \vartriangle ABC$ around the line $AP$ [img]https://cdn.artofproblemsolving.com/attachments/c/a/65157e2d49d0d4f0f087f3732c75d96a49036d.png[/img] p4. A class farewell event is attended by $10$ male students and $ 12$ female students. Homeroom teacher from the class provides six prizes to randomly selected students. Gifts that provided are one school bag, two novels, and three calculators. If the total students The number of male students who received prizes was equal to the total number of female students who received prizes. How many possible arrangements are there of the student who gets the prize? p5. It is known that $S =\{1945, 1946, 1947, ..., 2016, 2017\}$. If $A = \{a, b, c, d, e\}$ a subset of $S$ where $a + b + c + d + e$ is divisible by $5$, find the number of possible $A$'s.

Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

In the diagram, $ABCDEF$ is a regular hexagon with side length 2. Points $E$ and $F$ are on the $x$ axis and points $A$, $B$, $C$, and $D$ lie on a parabola. What is the distance between the two $x$ intercepts of the parabola? [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.3215445204635294, xmax = 7.383669550094284, ymin = -4.983460515387094, ymax = 6.688676116382409; pen zzttqq = rgb(0.6,0.2,0); pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((2,0)--(4,0)--(5,1.7320508075688774)--(4,3.4641016151377553)--(2,3.4641016151377557)--(1,1.732050807568879)--cycle, linewidth(1)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true); draw((2,0)--(4,0), linewidth(1)); draw((4,0)--(5,1.7320508075688774), linewidth(1)); draw((5,1.7320508075688774)--(4,3.4641016151377553), linewidth(1)); draw((4,3.4641016151377553)--(2,3.4641016151377557), linewidth(1)); draw((2,3.4641016151377557)--(1,1.732050807568879), linewidth(1)); draw((1,1.732050807568879)--(2,0), linewidth(1)); real f1 (real x) {return -0.58*x^(2)+3.46*x-1.15;} draw(graph(f1,-3.3115445204635297,7.373669550094284), linewidth(1)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /*yes i used geogebra fight me*/ [/asy]

Consider $n$ disks $C_{1}; C_{2}; ... ; C_{n}$ in a plane such that for each $1 \leq i < n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the [i]score[/i] of such an arrangement of $n$ disks to be the number of pairs $(i; j )$ for which $C_{i}$ properly contains $C_{j}$ . Determine the maximum possible score.

Given a convex polyhedron $F$. Its vertex $A$ has degree $5$, other vertices have degree $3$. A colouring of edges of $F$ is called nice, if for any vertex except $A$ all three edges from it have different colours. It appears that the number of nice colourings is not divisible by $5$. Prove that there is a nice colouring, in which some three consecutive edges from $A$ are coloured the same way. [i]D. Karpov[/i]

Calculate the limit $$\lim_{n \to \infty} \frac{1}{n} \left(\frac{1}{n^k} +\frac{2^k}{n^k} +....+\frac{(n-1)^k}{n^k} +\frac{n^k}{n^k}\right).$$ (For the calculation of the limit, the integral construction procedure can be followed).

Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.

Let there be $A=1^{a_1}2^{a_2}\dots100^{a_100}$ and $B=1^{b_1}2^{b_2}\dots100^{b_100}$ , where $a_i , b_i \in N$ , $a_i + b_i = 101 - i$ , ($i= 1,2,\dots,100$). Find the last 1124 digits of $P = A * B$.

The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza? $\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$

A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.

Find the number of ways to tile a $2 \times 2 \times 2 \times 2$ four dimensional hypercube with $2 \times 1 \times 1 \times 1$ blocks, with reflections and rotations of the large hypercube distinct.

Does there exist a natural number that can be represented as the product of two numeric palindromes in more than $100{}$ ways?

Let \[a=\dfrac{1^2}1+\dfrac{2^2}3+\dfrac{3^2}5+\cdots+\dfrac{1001^2}{2001}\] and \[b=\dfrac{1^2}3+\dfrac{2^2}5+\dfrac{3^2}7+\cdots+\dfrac{1001^2}{2003}.\] Find the integer closest to $a-b$. $\textbf{(A) }500\qquad\textbf{(B) }501\qquad\textbf{(C) }999\qquad\textbf{(D) }1000\qquad\textbf{(E) }1001$

Let $S$ be a set of nonnegative integers such that [list] [*] there exist two elements $a$ and $b$ in $S$ such that $a,b>1$ and $\gcd(a,b)=1$; and [*] for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $xy$ and the remainder when $x$ is divided by $y$ are in $S$. [/list] Prove that $S$ contains every nonnegative integer. [i]Jacob Paltrowitz[/i]

Four couples are to be seated in a row. If it is required that each woman may only sit next to her husband or another woman, how many different possible seating arrangements are there?

$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

Let $x_1,\ldots,x_n$ be given real numbers with $0<m\le x_i\le M$ for each $i\in\{1,\ldots,n\}$. Let $X$ be the discrete random variable uniformly distributed on $\{x_1,\ldots,x_n\}$. The mean $\mu$ and the variance $\sigma^2$ of $X$ are defined as $$\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.$$ By $X^2$ denote the discrete random variable uniformly distributed on $\{x_1^2,\ldots,x_n^2\}$. Prove that $$\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).$$