For any function $f:\mathbb{N}\to\mathbb{N}$ we define $P(n)=f(1)f(2)...f(n)$ . Find all functions $f:\mathbb{N}\to\mathbb{N}$ st for each $a,b$ : $$P(a)+P(b) | a! + b!$$

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

a) Does there exist a real number $x$ such that $x+\sqrt{3}$ and $x^2+\sqrt{3}$ are both rationals? b) Does there exist a real number $y$ such that $y+\sqrt{3}$ and $y^3+\sqrt{3}$ are both rationals?

The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.

Circle $C_1$ has center $O$ and radius $OA$, and circle $C_2$ has diameter $OA$. $AB$ is a chord of circle $C_1$ and $BD$ may be constructed with $D$ on $OA$ such that $BD$ and $OA$ are perpendicular. Let $C$ be the point where $C_2$ and $BD$ intersect. If $AC = 1$, find $AB$.

Let triangle$ABC(AB<AC)$ with incenter $I$ circumscribed in $\odot O$. Let $M,N$ be midpoint of arc $\widehat{BAC}$ and $\widehat{BC}$, respectively. $D$ lies on $\odot O$ so that $AD//BC$, and $E$ is tangency point of $A$-excircle of $\bigtriangleup ABC$. Point $F$ is in $\bigtriangleup ABC$ so that $FI//BC$ and $\angle BAF=\angle EAC$. Extend $NF$ to meet $\odot O$ at $G$, and extend $AG$ to meet line $IF$ at L. Let line $AF$ and $DI$ meet at $K$. Proof that $ML\bot NK$.

Let $N$ be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when $N$ is divided by 1000.

Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum $$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum

The solution to inequality $\left|\frac{1}{\log_{\frac{1}{2}}x}+2\right|>\frac{3}{2}$ is________(express answer with a set).

Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac12\right]$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\ldots$ find $n_{12}$

Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive integers.

Given an $m\times n$ grid with unit squares coloured either black or white, a black square in the grid is [i]stranded [/i]if there is some square to its left in the same row that is white and there is some square above it in the same column that is white. Find a closed formula for the number of $2\times n$ grids with no stranded black square. Note that $n$ is any natural number and the formula must be in terms of $n$ with no other variables.

On a sphere with radius $1$, a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $, $ B $ and $ C $. Prove that all such possible $ ABC $ planes pass through fixed point, and find the maximum possible area of the triangle $ ABC $

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

Find all pairs of integers $m, n \geq 2$ such that $$n\mid 1+m^{3^n}+m^{2\cdot 3^n}.$$

Let two ants stand on the perimeter of a regular $2019$-gon of unit side length. One of them stands on a vertex and the other one is on the midpoint of the opposite side. They start walking along the perimeter at the same speed counterclockwise. The locus of their midpoints traces out a figure $P$ in the plane with $N$ corners. Let the area enclosed by the convex hull of $P$ be $\tfrac{A}{B}\tfrac{\sin^m\left(\tfrac{\pi}{4038}\right)}{\tan\left(\tfrac{\pi}{2019}\right)}$, where $A$ and $B$ are coprime positive integers, and $m$ is the smallest possible positive integer such that this formula holds. Find $A+B+m+N$. [i]Note:[/i] The [i]convex hull[/i] of a figure $P$ is the convex polygon of smallest area which contains $P$.

Let $a$ be a positive real number and let $b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$. Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.

Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.

Let $ S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) $, where the $ b_j$ are arbitrary and the $ z_j$ are nonzero complex numbers . Prove that \[ |S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.\] [i]G. Halasz[/i]

$a,b,c,t$ are antural numbers and $k=c^{t}$ and $n=a^{k}-b^{k}$. a) Prove that if $k$ has at least $q$ different prime divisors, then $n$ has at least $qt$ different prime divisors. b)Prove that $\varphi(n)$ id divisible by $2^{\frac{t}{2}}$

[b]p1.[/b] Sarah needed a ride home to the farm from town. She telephoned for her father to come and get her with the pickup truck. Being eager to get home, she began walking toward the farm as soon as she hung up the phone. However, her father had to finish milking the cows, so could not leave to get her until fifteen minutes after she called. He drove rapidly to make up for lost time. They met on the road, turned right around and drove back to the farm at two-thirds of the speed her father drove coming. They got to the farm two hours after she had called. She walked and he drove both ways at constant rates of speed. How many minutes did she spend walking? [b]p2.[/b] Let $A = (a,b)$ be any point in a coordinate plane distinct from the origin $O$. Let $M$ be the midpoint of $OA$, and let $P$ be a point such that $MP$ is perpendicular to $OA$ and the lengths $\overline{MP}$ and $\overline{OM}$ are equal. Determine the coordinates $(x,y)$ of $P$ in terms of $a$ and $b$. Give all possible solutions. [b]p3.[/b] Determine the exact sum of the series $$\frac{1}{1 \cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \frac{1}{3\cdot 4\cdot 5} + ... + \frac{1}{98\cdot 99\cdot 100}$$ [b]p4.[/b] A six pound weight is attached to a four foot nylon cord that is looped over two pegs in the manner shown in the drawing. At $B$ the cord passes through a small loop in its end. The two pegs $A$ and $C$ are one foot apart and are on the same level. When the weight is released the system obtains an equilibrium position. Determine angle $ABC$ for this equilibrium position, and verify your answer. (Your verification should assume that friction and the weight of the cord are both negligible, and that the tension throughout the cord is a constant six pounds.) [img]https://cdn.artofproblemsolving.com/attachments/a/1/620c59e678185f01ca8743c39423234d5ba04d.png[/img] [b]p5.[/b] The four corners of a rectangle have the property that when they are taken three at a time, they determine triangles all of which have the same perimeter. We will consider whether a set of five points can have this property. Let $S = \{p_1, p_2, p_3, p_4, p_5\}$ be a set of five points. For each $i$ and $j$, let $d_{ij}$ denote the distance from $p_i$ to $p_j$. Suppose that $S$ has the property that all triangles with vertices in $S$ have the same perimeter. (a) Prove that $d$ must be the same for every pair $(i,j)$ with $i \ne j$. (b) Can such a five-element set be found in three dimensional space? Justify your answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered? [asy] size(120); draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1)); draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4));[/asy] $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100 $

Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\ $\textbf{(a)}$ Prove that $G$ is cyclic. \\ $\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?

$(a_i),(b_i)$ are nonconstant arithmetic and geometric progressions. $a_1=b_1,a_2/b_2=2,a_4/b_4=8$ Find $a_3/b_3$.

Solve the equation \[ \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ \] for $x$ where $0 < x < 90$.