We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be [i]balanced[/i] if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum. $a)$ Prove that if there exist [i]balanced[/i] sets of sizes $k,j$ then also a [i]balanced[/i] set of size $k+j-1$. $b)$ Prove that for all [i]odd[/i] $n\geq 7$ there exist a [i]balanced[/i] set of size $n$.

Prove that a)$\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ (mod $2017^2$ ) b)$\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ (mod $2017^2$ )

Let $a>0$, the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$, if for any positive reals $x$ and $y$, there is \[f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)\] then prove that $f(x)$ is a constant.

a) Prove that if $n$ is an odd perfect number then $n$ has the following form \[ n=p^sm^2 \] where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$. b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number

For each $n \ge 2$ define the polynomial $$f_n(x)=x^n-x^{n-1}-\dots-x-1.$$ Prove that (a) For each $n \ge 2$, $f_n(x)=0$ has a unique positive real root $\alpha_n$; (b) $(\alpha_n)_n$ is a strictly increasing sequence; (c) $\lim_{n \rightarrow \infty} \alpha_n=2.$

$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.

The perimeter of triangle $APM$ is $152,$ and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}.$ Given that $OP=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Find the least positive integer $ a$ such that $ 2001$ divides $ 55^n\plus{}a \cdot 32^n$ for some odd $ n$.

Let $f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4)$ where $a, b, c, d, e$ are integers, $a \not= 0$. Show that if $r_1 + r_2$ is a rational number, and if $r_1 + r_2 \neq r_3 + r_4$, then $r_1r_2$ is a rational number.

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.

Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N \equal{} AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that \[ \frac {NX} {NY} \equal{} \frac {AC} {AB}. \] [i]Cosmin Pohoata[/i]

Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

A fisherman has caught a number of fish. The three heaviest together make up $35\%$ of the total weight of the catch. He sells them. After that, the three lightest make up together $5/13$ of the weight of the rest. How many fish did he catch?

On the sides $AB$,$BC$, and $CA$ of $\triangle ABC$ are chosen points $C_1$, $A_1$, and $B_1$ respectively, in such way that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $X$. If $\angle A_1C_1B = \angle B_1C_1A$, prove that $CC_1$ is perpendicular to $AB$.

Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules: - If the square is green, write the number of red friends plus twice the number of blue friends. - If the square is red, write the number of blue friends plus twice the number of green friends. - If the square is blue, write the number of green friends plus twice the number of red friends. Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Knowing that $23$ October $1948$ was a Saturday, which is more frequent for New Year’s Day, Sunday or Monday?

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $R$ be the radius of circumcircle of $\triangle ABC$. Let $A',B',C'$ be the points on $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ respectively such that $AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2$. Prove that $O$ is incenter of $\triangle A'B'C'$.

Boris is driving on a remote highway. His car’s odometer reads $24942\text{ km}$, which Boris notices is a palindromic number, meaning it is not changed when it is reversed. “Hm,” he thinks, “it should be a long time before I see that again.” But it takes only $1$ hour for the odometer to once again show a palindromic number! How fast is Boris driving in $\text{km/h}$?

Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.