Let $n$ be a positive integer such that the sum of all its positive divisors (inclusive of $n$) equals to $2n + 1$. Prove that $n$ is an odd perfect square. related: https://artofproblemsolving.com/community/c6h515011 https://artofproblemsolving.com/community/c6h108341 (Putnam 1976) https://artofproblemsolving.com/community/c6h368488 https://artofproblemsolving.com/community/c6h445330 https://artofproblemsolving.com/community/c6h378928

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

On the side $AB$ of the non-isosceles triangle $ABC$, let the points $P$ and $Q$ be so that $AC = AP$ and $BC = BQ$. The perpendicular bisector of the segment $PQ$ intersects the angle bisector of the $\angle C$ at the point $R$ (inside the triangle). Prove that $\angle ACB + \angle PRQ = 180^o$.

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

The city of Neverreturn has $N$ bus stops numbered $1, 2, \cdots , N.$ Each bus route is one-way and has only two stops, the beginning and the end. The route network is such that departing from any stop one cannot return to it using city buses. When the mayor notices a route going from a stop with a greater number to a stop with a lesser number, he orders to exchange the number plates of its beginning and its end. Can the plate changing go on infinitely? [i](K. Ivanov )[/i]

Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \] Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.

Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$

In the xy-plane, what is the length of the shortest path from $(0, 0)$ to $(12, 16)$ that does not go inside the circle $(x - 6)^2 + (y - 8)^2 = 25$? $\text{(A)}\ 10\sqrt 3 \qquad \text{(B)}\ 10\sqrt 5 \qquad \text{(C)}\ 10\sqrt 3 + \frac{ 5\pi}{3} \qquad \text{(D)}\ 40\frac{\sqrt{3}}3 \qquad \text{(E)}\ 10+5\pi$

$\tfrac11+\tfrac13+\tfrac15=\tfrac12+\tfrac14+\tfrac16+\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 3 \qquad\textbf{(C)}\ 2: 5 \qquad\textbf{(D)}\ 3: 4 \qquad\textbf{(E)}\ 3: 5$

Determine the maximum value of the sum \[ S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}} \] over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying \[ \sum_{k=1}^{\infty}a_k=1. \]

Let $n\geq 2$ be a fixed positive integer. Let $\{a_1,a_2,...,a_n\}$ be fixed positive integers whose sum is $2n-1$. Denote by $S_{\mathbb{A}}$ the sum of elements of a set $A$. Find the minimal and maximal value of $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are two sets with the property that $\mathbb{X}\cup \mathbb{Y}=\{a_1,a_2,...,a_n\}$ and $\mathbb{X}\cap \mathbb{Y}=\emptyset.$ [i]Note: $\mathbb{X}$ and $\mathbb{Y}$ can have multiple equal elements. For example, when $n=5$ and $a_1=...=a_4=1$ and $a_5=5$, we can consider $\mathbb{X}=\{1,1,1\}$ and $\mathbb{Y}=\{1,5\}$. Moreover, in this case, $S_\mathbb{X}=3$ and $S_{\mathbb{Y}}=6.$[/i]

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$.

Let $a_n = n(2n+1)$. Evaluate \[ \biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |. \]

Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it. P.S. Proposed by yours truly :-D

Let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate \[\lim_{m \rightarrow \infty}\frac{\sum_{n = 1}^m \phi(60n)}{\sum_{n = 1}^m \phi(n)}\]

A whole number is written on the board. Its last digit is remembered is then erased and multiplied by $5$ added to the number that remained on the board after erasing. The number was originally written $7^{1998}$. After applying several such operations, can one get the number $1998^7$?

We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?

In triangle $ ABC$, $ AB\equal{}AC$, $ \angle A\equal{}40^\circ$. Point $ O$ is within the triangle with $ \angle OBC \cong \angle OCA$. The number of degrees in angle $ BOC$ is: $ \textbf{(A)}\ 110 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 70$