Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^n-1.$ If $s=2023$ (in base ten), compute $n$ (in base ten).

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying \[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95) \] for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$

A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns. The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.

Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

Is it true that there exist 100 lines in the plane, no three concurrent, such that they intersect in exactly 2002 points?

Let $n > 2$ and $n$ lamps numbered $1, 2, ..., n$ be connected in cyclic order: $1$ to $2, 2$ to $3, ..., n-1$ to $n, n$ to $1$. At the beginning all lamps are off. If the switch of a lamp is operated, the lamp and its $2$ neighbors change status: off to on, on to off. Prove that if $3$ does not divide $n$, then (all the) $2^n$ configurations can be reached and if $3$ divides $n$, then $2^{n-2}$ configurations can be reached.

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

Let $p(n)$ be the greatest odd divisor of $n$. Prove that \[\frac{1}{2^{n}}\sum_{k=1}^{2^{n}}\frac{p(k)}{k}> \frac{2}{3}.\]

Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

Suppose that $a=2^b$, where $b=2^{10n+1}$. Prove that $a$ is divisible by 23 for any positive integer $n$

Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions: (a) $k \le 4r$ (b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $\ell$ such that $(m - 1)! < \ell < (m + 1)! + 1$

A circle inscribed within quadrilateral $ABCD$ is tangent to $AB$ at $E$, to $BC$ at $F$, to $CD$ at $G$, and to $DA$ at $H$. Suppose that $AE = 6$, $EB = 30$, $CG = 10$, and $GD = 2$. Compute $EF^2 + F G^2 + GH^2 + HE^2$. .

$p,q$ are nonnegative integers.Given two conditions: A: $p^3-q^3$ is an even number. B: $p+q$ is an even number. Then, which one of the followings are true? $(\text{A})$A is sufficient but unnecessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient and necessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

A small chunk of ice falls from rest down a frictionless parabolic ice sheet shown in the figure. At the point labeled $\mathbf{A}$ in the diagram, the ice sheet becomes a steady, rough incline of angle $30^\circ$ with respect to the horizontal and friction coefficient $\mu_k$. This incline is of length $\frac{3}{2}h$ and ends at a cliff. The chunk of ice comes to a rest precisely at the end of the incline. What is the coefficient of friction $\mu_k$? [asy] size(200); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(sqrt(3),0)--(0,1)); draw(anglemark((0,1),(sqrt(3),0),(0,0))); label("$30^\circ$",(1.5,0.03),NW); label("A", (0,1),NE); dot((0,1)); label("rough incline",(0.4,0.4)); draw((0.4,0.5)--(0.5,0.6),EndArrow); dot((-0.2,4/3)); label("parabolic ice sheet",(0.6,4/3)); draw((0.05,1.3)--(-0.05,1.2),EndArrow); label("ice chunk",(-0.5,1.6)); draw((-0.3,1.5)--(-0.25,1.4),EndArrow); draw((-0.2,4/3)--(-0.19, 1.30083)--(-0.18,1.27)--(-0.17,1.240833)--(-0.16,1.21333)--(-0.15,1.1875)--(-0.14,1.16333)--(-0.13,1.140833)--(-0.12,1.12)--(-0.11,1.100833)--(-0.10,1.08333)--(-0.09,1.0675)--(-0.08,1.05333)--(-0.07,1.040833)--(-0.06,1.03)--(-0.05,1.020833)--(-0.04,1.01333)--(-0.03,1.0075)--(-0.02,1.00333)--(-0.01,1.000833)--(0,1)); draw((-0.6,0)--(-0.6,4/3),dashed,EndArrow,BeginArrow); label("$h$",(-0.6,2/3),W); draw((0.2,1.2)--(sqrt(3)+0.2,0.2),dashed,EndArrow,BeginArrow); label("$\frac{3}{2}h$",(sqrt(3)/2+0.2,0.7),NE); [/asy] $ \textbf{(A)}\ 0.866\qquad\textbf{(B)}\ 0.770\qquad\textbf{(C)}\ 0.667\qquad\textbf{(D)}\ 0.385\qquad\textbf{(E)}\ 0.333 $

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.

Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by James Lin[/i]