Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

Find all pairs $(x, y)$ of real numbers satisfying the equations \begin{align*} x^2+y&=xy^2 \\ 2x^2y+y^2&=x+y+3xy. \end{align*}

Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies: (i) $f(1995) = 1996$ (ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$. Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .

Prove or disprove: $\exists n\in N$ , s.t. $324 + 455^n$ is prime.

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

For each integer $n\geq0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2\leq n$. Define a sequence by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k\geq0$. For what positive integers $A$ is this sequence eventually constant?

Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$.

The board was placed $$ \begin{array}{rcl}<br /> 1 & = & 1 \\<br /> 2 + 3 + 4 & = & 1 + 8 \\<br /> 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\<br /> 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\<br /> & \ldots &<br /> \end{array}$$ Write such a formula for the $ n $-th row of the array that, with the substitutions $ n = 1, 2, 3, 4 $, would give the above four lines of the array and would be true for every natural $ n $.

A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset

Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same; 3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same. Find all $ m,n$ such that there exists a good $ m\times n$ board.

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.

In a unit cube $ABCD - EFGH$, an equilateral triangle $BDG$ cuts out a circle from the circumsphere of the cube. Find the area of the circle.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle . (A. Andjans, Riga)

Let $ \mathbb{K} $ be a field and let be a matrix $ M\in\mathcal{M}_3(\mathbb{K} ) $ having the property that $ \text{tr} (A) =\text{tr} (A^2) =0 . $ Show that there is a $ \mu\in \mathbb{K} $ such that $ A^3=\mu A $ or $ A^3=\mu I. $ [i]Cristinel Mortici[/i]

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Three congruent equilateral triangles $T_1$, $T_2$, and $T_3$ are stacked from left to right inside rectangle $JHMT$ such that the bottom left vertex of $T_1$ is $T$, the bottom side of $T_1$ lies on $\overline{MT}$, the bottom left vertex of $T_2$ is the midpoint of a side of $T_1$, the bottom left vertex of $T_3$ is the midpoint of a side of $T_2$, and the other two vertices of $T_3$ lie on $\overline{JH}$ and $\overline{HM}$, as shown below. Given that rectangle $JHMT$ has area $2022$, find the area of any one of the triangles $T_1$, $T_2$, or $T_3$. [asy] unitsize(0.111111111111111111cm); real s = sqrt(4044/sqrt(75)); real l = 5s/2; real w = s * sqrt(3); pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3; J = (0,w); H = (l,w); M = (l,0); T = (0,0); V1 = (s,0); V2 = (s/2,s * sqrt(3)/2); V3 = (V1+V2)/2; V4 = (3 * s/4+s,s * sqrt(3)/4); V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2); V6 = (V4+V5)/2; V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4); V8 = (l-s/2,w); C1 = (T+V1+V2)/3; C2 = (V3+V4+V5)/3; C3 = (V6+V7+V8)/3; draw(J--H--M--T--cycle); draw(V1--V2--T); draw(V3--V4--V5--cycle); draw(V6--V7--V8--cycle); label("$J$", J, NW); label("$H$", H, NE); label("$M$", M, SE); label("$T$", T, SW); label("$T_1$", C1); label("$T_2$", C2); label("$T_3$", C3); [/asy]

Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$. Show that for any $k$, $|S(k)|\leq n-1$.