Let $\triangle ABC$ be a triangle. Let points $D$ and $E$ be on segment $BC$ in the order $B, D, E, C$ such that $\angle BAD =$ $\angle DAE = \angle EAC.$ Suppose also that $BD = F_{2024}, DE = F_{2025}, EC = F_{2027},$ where $F_k$ is the $k$th Fibonacci number where $F_1 = F_2 = 1.$ To the nearest degree, $\angle BAC$ is $n^\circ.$ Find $n.$

Prove that every rational number is representable as $x^4+y^4-z^4-t^4$ with rational $x,y,z,t$.

Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.

Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$

Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$ Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$

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]