Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$. Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.

Anna must write $7$ positive integers, not necessarily distinct, around a circle such that the following conditions are met: $\bullet$ The sum of the seven numbers equals $36$. $\bullet$ If two numbers are neighbours, the difference between the largest and the smallest is equal to $2$ or $3$. Find the maximum value of the largest of the numbers that Anna can write.

Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$.

Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$

Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and$$t_n = \frac{5t_{n-1}+1}{25t_{n-2}}$$for all $n \ge 3$. Then $t_{2020}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

Find all integers $n \ge 3$ for which the polynomial \[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\] can be written as a product of two non-constant polynomials with integer coefficients.

Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$

Point $D$ is the midpoint of $BC$, where $ABC$ is an isosceles triangle ($AB=AC$). On circle $(ABD)$ a point $P \neq A$ is chosen. $O$ is the circumcenter of $ACP$, $Q$ is the foot of the perpendicular from $C$ onto $AO$. Prove that the circumcenter of triangle $ABQ$ lies on the line $AP$

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*} Calculate $f(14,52)$.

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.

Given two points $ A$ and $ B$ and two circles, $\Gamma_1$ with center $A$ and passing through $ B$, and $\Gamma_2$ with center $ B$ and passing through $ A$. Line $AB$ meets $\Gamma_2$ at point $C$. Point $D$ lies on $\Gamma_2$ such that $\angle CDB = 57^o$. Line $BD$ meets $\Gamma_1$ at point $E$. What is $\angle CAE$, in degrees?

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.

We say that a string of digits from $0$ to $9$ is valid if the following conditions hold: First, for $2 \le k \le 4$, no consecutive run of $k$ digits sums to a multiple of $10$. Second, between any two $0$s, there are at least $3$ other digits. Find the last four digits of the number of valid strings of length $2020$.

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain. [i]Proposed by Merlijn Staps, The Netherlands[/i]

Given is a triangle $ABC$ with incircle $\omega$, tangent to $BC, CA, AB$ at $D, E, F$. The perpendicular from $B$ to $BC$ meets $EF$ at $M$, and the perpendicular from $C$ to $BC$ meets $EF$ at $N$. Let $DM$ and $DN$ meet $\omega$ at $P$ and $Q$. Prove that $DP=DQ$.

A subset of $\{1, 2, 3, ... ... , 2015\}$ is called good if the following condition is fulfilled: for any element $x$ of the subset, the sum of all the other elements in the subset has the same last digit as $x$. For example, $\{10, 20, 30\}$ is a good subset since $10$ has the same last digit as $20 + 30 = 50$, $20$ has the same last digit as $10 + 30 = 40$, and $30$ has the same last digit as $10 + 20 = 30$. (a) Find an example of a good subset with 400 elements. (b) Prove that there is no good subset with 405 elements.

If $a^2+b^2+c^2+d^2=1$, prove that $$(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4$$ When does equality holds?

Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}$, $|AB|=2$, $|AD|=4$ and $|BC|=|BD|$. If the line through $C$ which is parallel to $AB$ meets $AD$ at $E$, what is $|CE|$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it covers a total area of $\frac{23}{3}$. [asy] fill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey); fill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey); fill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey); fill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey); draw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4)); draw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4)); draw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4)); draw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4)); draw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4)); draw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4)); [/asy] Find, with proof, the minimum $n$ such that the figure covers an area of at least $\sqrt{63}$.