Which of these numbers is less than its reciprocal? $\textbf{(A)}\ -2\qquad \textbf{(B)}\ -1\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2$

Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$, \[P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).\]

Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$. Show that $\widehat{PAC} = 2 \widehat{CPA}.$

Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur? $ \textbf{(A) } \text{Tuesday} \qquad \textbf{(B) } \text{Wednesday} \qquad \textbf{(C) } \text{Thursday} \qquad \textbf{(D) } \text{Friday} \qquad \textbf{(E) } \text{Saturday}$

Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. [i]Proposed by James Rickards, Canada[/i]

All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? $ \textbf{(A)}\ 2\text{ by }4 \qquad\textbf{(B)}\ 2\text{ by }6 \qquad\textbf{(C)}\ 2\text{ by }8 \qquad\textbf{(D)}\ 4\text{ by }4 \qquad\textbf{(E)}\ 4\text{ by }8 $

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.

Prova that any integer $ Z $ has a unique representation $$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$ where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $

Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.

A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$, $K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On the line $OB$ a point $L$ is taken such that the lines $OA$ and $KL$ are parallel. Let $M$ be the intersection point of the circle $\omega$ circumscribed around triangle $KLB$, with line $AK$, with $M$ different from $K$. Prove that line $OM$ touches circle $\omega$.

Prove that the number $$\bigg\lfloor\frac{2024}{1}\bigg\rfloor+\bigg\lfloor\frac{2023}{2}\bigg\rfloor+\bigg\lfloor\frac{2022}{3}\bigg\rfloor+\dots +\bigg\lfloor\frac{1013}{1012}\bigg\rfloor$$ is even.

Determine the integers $a, b, c$ for which $$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$

Let $a_1, a_2, \dots, a_n$ be positive integers such that none of them is a multiple of $5$. What is the largest integer $n<2005$, such that $a_1^4 + a_2^4 + \cdots + a_n^4$ is divisible by $5$? $ \textbf{(A)}\ 2000 \qquad\textbf{(B)}\ 2001 \qquad\textbf{(C)}\ 2002 \qquad\textbf{(D)}\ 2003 \qquad\textbf{(E)}\ 2004 $

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$. Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$, there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$. Suppose that the minimum possible value of $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. (A [i]lattice point[/i] is a point with all integer coordinates.) [hide="Clarifications"] [list] [*] $k = 2013$, i.e. the problem should read, ``... there exists at least one index $i$ such that $1\le i\le 2013$ ...''. An earlier version of the test read $1 \le i \le k$.[/list][/hide] [i]Anderson Wang[/i]

What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$? $\text{(A)}\ -24 \qquad \text{(B)}\ -3 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.

In the plane, we have any polygon that does not intersect itself and is closed. Given a point that is not on the edge of the polygon. How can we determine whether it is inside or outside the polygon? (the polygon has a finite number of sides) [hide=original wording]En el plano se tiene un poligono cualquiera que no se corta a si mismo y que es cerrado. Dado un punto que no esta sobre el borde del poligono, Como determinara se esta dentro o fuera del poligono? (el poligono tiene un numero nito de lados)[/hide]

In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality: $A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$

The $4$ code words $$\square * \otimes \,\,\,\, \oplus \rhd \bullet \,\,\,\, * \square \bullet \,\,\,\, \otimes \oslash \oplus$$ they are in some order $$AMO \,\,\,\, SUR \,\,\,\, REO \,\,\,\, MAS$$ Decrypt $$\otimes \oslash \square * \oplus \rhd \square \bullet \otimes $$