Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits?

Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.

Let $f_n$ be $n$th Fibonacci number defined by recurrence $f_{n+1} - f_n - f_{n-1} = 0$, $n \in \mathbb{N}$ and initial conditions $f_0 = 0$, $f_1 = 1$. Prove that for any $n \in \mathbb{N}$ $$(n - 1) (n + 1) (2nf_{n+1} - (n + 6) f_n)$$is divisible by 150 for any $n \in \mathbb{N}$.

Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \][i]Proposed by David Stoner[/i]

Let $\mathcal{P}$ be a convex polygon and $\textbf{T}$ be a triangle with vertices among the vertices of $\mathcal{P}$. By removing $\textbf{T}$ from $\mathcal{P}$, we end up with $0, 1, 2,$ or $3$ smaller polygons (possibly with shared vertices) which we call the effect of $\textbf{T}$. A triangulation of $P$ is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of $\mathcal{P}$ $\underline{\text{beautiful}}$, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of $\mathcal{P}$ is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?

It is initially considered a number of three different digits, none of which is equal to zero. Changing instead two of its digits meet a second number less than the first. If the difference between the first and second is a two-digit number and the sum of the first and the second is a palindromic number less than $500$, what are the palindromics that can be obtained?

A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false. I. The digit is 1. II. The digit is not 2. III. The digit is 3. IV. The digit is not 4. Which one of the following must necessarily be correct? $ \textbf{(A)}\ \text{I is true.} \qquad \textbf{(B)}\ \text{I is false.}\qquad \textbf{(C)}\ \text{II is true.} \qquad \textbf{(D)}\ \text{III is true.} \qquad \textbf{(E)}\ \text{IV is false.}$

Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/b/b/34883cf73da568ca237a13fbc2e0fb9322c2e5.png[/img]

Find the angles of a convex quadrilateral $ABCD$ such that $\angle ABD = 29^o$, $\angle ADB = 41^o$, $\angle ACB = 82^o$ and $\angle ACD = 58^o$

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9:1, and the ratio of blue to green marbles in Jar 2 is 8:1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2? $\textbf{(A) } 5 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 25 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 50$

In an isosceles triangle $ABC$ ($AB = AC$), the bisector of the angle $B$ intersects $AC$ at point $D$ such that $BC = BD + AD$. Find $\angle A$.

Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true: a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row. b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.

Does there exist a real sequence $\{a_n\}_{n=1}^\infty$ such that $$a_na_{n+1}\ge a_{n+2}^2 +1$$ for all $n\ge 1$?

Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. [i]M. Kungodjin[/i]

Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$. (Patrik Bak, Michal Rolínek)

Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n. $ Prove that there exists a real number $ a $ such that $ a+a_1,a+a_2,\ldots ,a+a_n $ are all irrational.

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

A twelve foot tree casts a five foot shadow. How long is Henry’s shadow (at the same time of day) if he is five and a half feet tall?