Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.

Let $N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1$. Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.

For any positive integer $n$ and real numbers $a_i>0$ ($i=1,2,...,n$), prove that \[\sum_{k=1}^n \frac{k}{a_1^{-1}+a_2^{-1}+...+a_k^{-1}}\leq 2\sum_{k=1}^n a_k.\] Discuss if the "$2$" at the right hand side of the inequality can or cannot be replaced by a smaller real number.

Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

A sphere rolls along two intersecting straight lines. Find the locus of its center.

2. Positive integer will be called white, if it is equal to $1$ or is a product of even number of primes (not necessarily distinct). Rest of the positive integers will be called black. Determine whether there exists a positive integer which sum of white divisors is equal to sum of black divisors

Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\] Prove that \[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]

The following is the construction of the [I]twindragon[/I] fractal. $I_0$ is the solid square region with vertices $(0,0),$ $ (\tfrac{1}{2}, \tfrac{1}{2}),$ $(1,0), (\tfrac{1}{2}, -\tfrac{1}{2}).$ Recursively, the region $I_{n+1}$ is consists of two copies of $I_n:$ one copy which is rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\tfrac{1}{\sqrt{2}},$ and another copy which is also rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\tfrac{1}{\sqrt{2}}$ and translated by $(\tfrac{1}{2}, -\tfrac{1}{2}).$ We have displayed $I_0$ and $I_1$ below. Let $I_{\infty}$ be the limiting region of $I_0, I_1, \ldots.$ The area of the smallest convex polygon which encloses $I_{\infty}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Find $a+b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/b/c/50b5d5a70c293329cf693bfaef823fb2813b07.png[/img] [/center]

Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.

Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.

For a positive integer $n$, does there exist a permutation of all its positive integer divisors $(d_1 , d_2 , \ldots, d_k)$ such that the equation $d_kx^{k-1} + \ldots + d_2x + d_1 = 0$ has a rational root, if: a) $n = 2024$; b) $n = 2025$? [i]Proposed by Mykyta Kharin[/i]

Show that the set of positive integers which cannot be represented as a sum of distinct perfect squares is finite.

How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entires in each column is 0?

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

Given is a prime number $p$. Prove that the number $$p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!$$ is divisible by $$\prod_{i=1}^{p}(p^i)!.$$

Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$. Prove that $c(a-1)+b$ does not divide $c(b-1)+a$. [i]Proposed by Dominik Burek, Poland[/i]

A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. Prove that $\angle BAP = \angle P'AC$.

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that: [list] [*][b](i)[/b] mathematics was ranked among the most preferred courses by all students; [*][b](ii)[/b] no student ranked music among the least preferred ones; [*][b](iii) [/b]all students preferred history to geography and physics to biology; and [*][b](iv)[/b] no two rankings were the same. [/list] Find the greatest possible value for the number of students in this school.

Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$ [color = red]The official statement does not have the final period.[/color]

Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

For $k=1,2,\ldots ,2011$ we denote $S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}$. Compute the sum $S_1+S_1^2+S_2^2+\cdots +S_{2011}^2$.

Prove that $A=10^{n^3-n+2}$ can be written as a sum of four perfect cubes.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D