It is the year 2005 now. According to a legend there is a monster that awakes every now and then to swallow everyone who is solving this problem, and then falls back asleep for as many years as the sum of the digits of that year. The monster first hit AoPS in the year +234. Prove you're safe this year, as well as for the coming 10 years.

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

Let $S = \{1, 2, 3, 5, 6, 10, 15, 30\}$. For each of the $64$ ordered pairs $(a, b)$ of elements of $S$, AJ computes $gcd(a, b)$. They then sum all of the numbers they computed. What is AJ’s sum?

Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying $f(xy+c)=f(x)+f(y)$, for all $x,y \in S$

Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$

Let $K$ be a knot in the $3$-dimensional space (that is a differentiable injection of a circle into $\mathbb{R}^3$, and $D$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of $D$ in black and color the diagram $D$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define $\Gamma_B(D)$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them. [list=a] [*]Determine all knots having a diagram $D$ such that $\Gamma_B(D)$ has at most $3$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/*] [*]Prove that for any knot and any diagram $D$, $\Gamma_B(D)$ has an odd number of spanning trees.[/*] [/list]

Edward is trying to spell the word "CAT". He has an equal chance of spelling the word in any order of letters (i.e. TAC or TCA). What is the probability that he spells "CAT" incorrectly? $\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac13\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac23\qquad\textbf{(E) }\dfrac56$

Let $x_{1}$ and $x_{2}$ be relatively prime positive integers. For $n \ge 2$, define $x_{n+1}=x_{n}x_{n-1}+1$.[list=a][*] Prove that for every $i>1$, there exists $j>i$ such that ${x_{i}}^{i}$ divides ${x_{j}}^{j}$. [*] Is it true that $x_{1}$ must divide ${x_{j}}^{j}$ for some $j>1$? [/list]

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

Let \( f(x) = \frac{100^x}{100^x + 10} \). Determine the value of: \[ f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right) \]

Find all natural numbers $n$ such that when we multiply all divisors of $n$, we will obtain $10^9$. Prove that your number(s) $n$ works and that there are no other such numbers. ([i]Note[/i]: A natural number $n$ is a positive integer; i.e., $n$ is among the counting numbers $1, 2, 3, \dots$. A [i]divisor[/i] of $n$ is a natural number that divides $n$ without any remainder. For example, $5$ is a divisor of $30$ because $30 \div 5 = 6$; but $5$ is not a divisor of $47$ because $47 \div 5 = 9$ with remainder $2$. In this problem, we consider only positive integer numbers $n$ and positive integer divisors of $n$. Thus, for example, if we multiply all divisors of $6$ we will obtain $36$.)

Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

Find all pairs $(a,b)$ of non-negative integers such that $2017^a=b^6-32b+1$.

The natural numbers $a_1 <a_2 <.... <a_n\le 2n$ are such that the least common multiple of any two of them is greater than $2n$. Prove that $a_1 >\left[\frac{2n}{3}\right]$.

Express $\sqrt{7+4\sqrt{3}}+\sqrt{7-4\sqrt{3}}$ in the simplest possible form.

Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a} {b-c} \right | + \left | \dfrac {b} {c-a} \right | + \left | \dfrac {c} {a-b} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c$ positive, then all equality cases disappear, but the value $2$ remains the best constant possible. ([i]Dan Schwarz[/i])

Let $n$ be a natural number. The leader of the math team invites $n$ girls for winter training, and each leaves her two gloves in a common box upon entry. The mischievous little brother randomly pairs the gloves into pairs, where each pair consists of one left glove and one right glove. A pairing is called [i]weak[/i] if there is a set of $k < \frac{n}{2}$ pairs containing gloves of exactly $k$ girls. Find the probability that the pairing is not weak.

The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$. [i]Proposed by Jeff Lin[/i]

Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that \[ 1 < \frac ab < 1 +\frac 1n.\]

Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?

Simplify $ \left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$; the result is: $ \textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2$

Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]