All problems are text-only with mathematical notation in LaTeX. The following conventions are observed: - \\(\\verb|amssymb|\\), \\(\\verb|amsmath|\\) and \\(\\verb|enumitem|\\) packages are used. No other package is assumed. - Variable names are given in math mode e.g. \\(a\\), \\(x\\), \\(y\\). \\(\verb|\mathcal|\\) letters may be used, e.g. \\(\\mathcal{R}\\), \\(\\mathcal{S}\\). - Integers such as 1, 2 can be outside of math mode or inside. Large integers such as \\(10^{50}\\) are in math mode. - \\(\\verb|itemize|\\) and \\(\\verb|enumerate|\\) environments may be used (with the latter using the conventions from the \\(\\verb|enumitem|\\) package). - \\(\\mathtt{\\$}...\\mathtt{\\$}\\), \\(\\verb|\\ \(...\\ \)|\\), \\(\\verb|\\\[...\\\]|\\), \\(\\verb|\\begin{equation}...\\end{equation}|\\), and \\(\\verb|\\begin{equation\*}...\\end{equation\*}|\\) may all be used for equations. - No environments other than the ones mentioned above are used. - \\(\\verb|\\ldots|\\), \\(\\verb|\\cdots|\\), \\(\\verb|...|\\) can be used to continue a finite sequence or infinite. - Multiplication can be represented by simple adjacency \\(2x\\) or using LaTeX \\(\\verb|\\cdot|\\) command for \\(x \\cdot y\\) or LaTeX \\(\\verb|\\times|\\) command for \\(x \\times y\\). - \\(n!\\) for an integer \\(n \\geq 0\\) represents the factorial function \\(n! = n\\cdot(n-1)\\cdot(n-2) \\cdots 2 \\cdot 1\\) (with \\(0!=1\\)). - \\(\\log\\), \\(\\sin\\), \\(\\cos\\) are generally in math mode. Unless a base is explictly stated (e.g., $\log_{2}$), $\log$ refers to the natural logarithm. - Fractions may be represented using \\(\\frac{a}{b}\\) or inline using \\(a/b\\). - \\(\\lvert x \\rvert\\) represents the absolute value of \\(x\\), e.g., \\(\\lvert -2 \\rvert = 2\\). - \\(x \%\\), where \\(x\\) is a real number, represents the value \\(\\frac{x}{100}\\) - Standard geometric notation is used such as the triangle \\(\\triangle ABC\\) with sides \\(AB\\), \\(BC\\), and \\(AC\\). Standard notation is also used for angles, e.g., \\(\\angle ABC\\). Angles may be specified in degrees \\(\\angle ABC = 60^\\circ\\). Measures with no units should be assumed to be in radians. - Standard trigonometric operators including \\(\\sin\\), \\(\\cos\\), \\(\\tan\\) may be used. - Sets are denoted using standard curly bracket notation, e.g., \\(\\{1,2,3\\}\\). - \\(\\lfloor x \\rfloor\\) represents the integer part of the real number \\(x\\) ie the greatest integer less than or equal to \\(x\\). \\(\\lceil x \\rceil\\) represents the smallest integer greater than or equal to \\(x\\). \\(\\{x\\}\\) for a single value \\(x\\) refers to the fractional part of the real number \\(x\\) ie \\(x - \\lfloor x \\rfloor\\). - \\(\\verb|\\left|\\) and \\(\\verb|\\right|\\) may be used in conjunction with other LaTeX commands, e.g. \\(\\verb|\\left \\lceil \\frac{x}{2}\\right\\rceil|\\) producing \\(\\left \\lceil \\frac{x}{2}\\right\\rceil\\). - Overline notation \\(\\overline{abcd}\\) represents the integer formed by the digits \\(a\\),\\(b\\),\\(c\\),\\(d \\in \\{0, 1, \\ldots, 9\\}\\) with \\(a \\neq 0\\), e.g. with \\(a=1\\), \\(b=2\\): \\(\\overline{ab}=12\\) and \\(\\overline{bab}=212\\). - \\(\\binom{a}{b}\\) is interpreted as the binomial coefficient, representing the number of ways of choosing \\(b \\geq 0\\) elements from a set of size \\(a \\geq 0\\). We use the convention that \\(\\binom{0}{0}=1\\) and \\(\binom{a}{b}=0\\) if \\(b>a\\). - \\(\\mathbb{N},\\mathbb{Z},\\mathbb{Q}\\), and \\(\\mathbb{R}\\) are used to denote the set of positive integers, integers, rational numbers, and real numbers, respectively (where positive means \\(>0\\)). For an integer \\(n\\), the notation \\(\\mathbb{Z}_{\\geq n}\\) may be used to denote the set of integers \\(\\geq n\\). - Standard notation for functions will be used such as \\(f\colon S \to T\\) where \\(S\\) and \\(T\\) are sets. - We allow informal probability language such as: a point is chosen uniformly at random in the interval \\([0, 1]\\). - The sum over the empty set is \\(0\\), and the product over the empty set is \\(1\\). - For integers \\(l\\), \\(m\\), and \\(n\\), \\(l^{m^n}\\) denotes \\(l^{\\left(m^n\\right)}\\). - British or American versions of English can be used. Thus ''highest common factor'' means the same as ''greatest common divisor''. - A prefix or subscript may be used to indicate features of a triangle associated with vertices. Thus triangle \\(ABC\\) has three altitudes, and the one dropped from \\(A\\) could be denoted the altitude through \\(A\\), the \\(A\\)-altitude or the altitude \\(h_a\\). Similarly, for median lines. - \\(:=\\) means 'is defined to be equal to'. - Taxonomy is *Bourbakist*: equilateral triangles are isosceles triangles, squares are rectangles and rectangles are parallelograms. - The word *trapezium* can mean different things in different countries. For the purposes of this competition, a trapezium is a quadrilateral with at least one pair of parallel opposite sides. **Please read this section carefully as there have been changes from the answer format used for AIMO2. We recommend looking through the reference problems to see examples.** - All answers must be reported as integers in the range \\(0\\) to \\(99999\\) inclusive. - All questions are worded so that the final answer will lie in this range (meaning any modulus required is explicitly given in the problem statement). - For questions of the form: `What is the remainder when \\(N\\) is divided by \\(q\\)?' where \\(q\\) is a positive integer and \\(N\\) is an integer, you should return the unique integer \\(r\\) such that \\(0 \\leq r