[size=130][b]Higher Secondary: 2011[/b] [/size] Time: 4 Hours [b]Problem 1:[/b] Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709 [b]Problem 2:[/b] In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708 [b]Problem 3:[/b] $E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683 [b]Problem 4:[/b] Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707 [b]Problem 5:[/b] In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706 [b]Problem 6:[/b] $p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693 [b]Problem 7:[/b] Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ [b]friendship/enmity[/b]. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694 [b]Problem 8:[/b] $ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$ http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704 [b]Problem 9:[/b] The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703 [b]Problem 10:[/b] Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702 The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum[/url]: http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678

Find all prime numbers $p$ and $q$ such that $3p^4+5q^4+15=13p^2q^2$.

Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)

Define a finite sequence $ \left( s_i \right)_{1\le i\le 2004} $ with $ s_0+2=s_1+1=s_2=2 $ and the recurrence relation $$ s_n=1+s_{n-1} +s_{n-2} -s_{n-3} . $$ Calculate its last element.

For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers whose partial numbers are all prime. Justify your opinion.

Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$

Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the expected number of pennies showing heads?

Prove the following equality for all positive integers $m,n$: $$\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}$$

Let $n \leq 100$ be an integer. Hare puts real numbers in the cells of a $100 \times 100$ table. By asking Hare one question, Wolf can find out the sum of all numbers of a square $n \times n$, or the sum of all numbers of a rectangle $1 \times (n - 1)$ (or $(n - 1) \times 1$). Find the greatest $n{}$ such that, after several questions, Wolf can find the numbers in all cells, with guarantee.

In a tennis tournament where $ n$ players $ A_1,A_2,\dots,A_n$ take part, any two players play at most one match, and $ k \leq \frac {n(n \minus{} 1)}{2}$ $ 2$ matches are played. The winner of a match gets $ 1$ point while the loser gets $ 0$. Prove that a sequence $ d_1,d_2,\dots,d_n$ of nonnegative integers can be the sequence of scores of the players ($ d_i$ being the score of$ A_i$) if and only if $ (i)\ \ d_1 \plus{} d_2 \plus{} \dots \plus{} d_n \equal{} k$, and $ (ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}$, the number of matches between the players in $ X$ is at most $ \sum_{A_j\in X}d_j$

A point object of mass $m$ is connected to a cylinder of radius $R$ via a massless rope. At time $t = 0$ the object is moving with an initial velocity $v_0$ perpendicular to the rope, the rope has a length $L_0$, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds $T_{max}$. Express your answers in terms of $T_{max}$, $m$, $L_0$, $R$, and $v_0$. [asy] size(200); real L=6; filldraw(CR((0,0),1),gray(0.7),black); path P=nullpath; for(int t=0;t<370;++t) { pair X=dir(180-t)+(L-t/180)*dir(90-t); if(X.y>L) X=(X.x,L); P=P--X; } draw(P,dashed,EndArrow(size=7)); draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7)); filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the kinetic energy of the object at the instant that the rope breaks? $ \textbf{(A)}\ \frac{mv_0^2}{2} $ $ \textbf{(B)}\ \frac{mv_0^2R}{2L_0} $ $ \textbf{(C)}\ \frac{mv_0^2R^2}{2L_0^2} $ $ \textbf{(D)}\ \frac{mv_0^2L_0^2}{2R^2} $ $ \textbf{(E)}\ \text{none of the above} $

Prove the following equality: $4 sin\frac{2\pi }{7}-tg \frac{\pi }{7}=\sqrt{7}$

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

In the figure the sum of the distances $AD$ and $BD$ is [asy] draw((0,0)--(13,0)--(13,4)--(10,4)); draw((12.5,0)--(12.5,.5)--(13,.5)); draw((13,3.5)--(12.5,3.5)--(12.5,4)); label("A", (0,0), S); label("B", (13,0), SE); label("C", (13,4), NE); label("D", (10,4), N); label("13", (6.5,0), S); label("4", (13,2), E); label("3", (11.5,4), N); [/asy] $ \textbf{(A)}\ \text{between 10 and 11} \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ \text{between 15 and 16} \qquad\textbf{(D)}\ \text{between 16 and 17} \qquad\textbf{(E)}\ 17 $

The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

Let $ABC$ be a triangle with an obtuse angle at $A$. Let $E$ and $F$ be the intersections of the external bisector of angle $A$ with the altitudes of $ABC$ through $B$ and $C$ respectively. Let $M$ and $N$ be the points on the segments $EC$ and $FB$ respectively such that $\angle EMA = \angle BCA$ and $\angle ANF = \angle ABC$. Prove that the points $E, F, N, M$ lie on a circle.

Let $a,b,c$ be positive reals. Prove that \[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \] [i]Calvin Deng.[/i]

Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.

[u]Round 9[/u] [b]p25.[/b] Find the largest prime factor of $1031301$. [b]p26.[/b] Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $\angle ABC = 90^o$ , $AB = 5$, $BC = 20$, $CD = 15$. Let $X$, $Y$ be the intersection of the circle with diameter $BC$ and segment $AD$. Find the length of $XY$. [b]p27.[/b] A string consisting of $1$’s, $2$’s, and $3$’s is said to be a superpermutation of the string $123$ if it contains every permutation of $123$ as a contiguous substring. Find the smallest possible length of such a superpermutation. [u]Round 10[/u] [b]p28.[/b] Suppose that we have a function $f (x) = x^3 -3x^2 +3x$, and for all $n \ge 1$, $f^n(x)$ is defined by the function $f$ applied $n$ times to $x$. Find the remainder when $f^5(2019)$ is divided by $100$. [b]p29.[/b] A function $f : {1,2, . . . ,10} \to {1,2, . . . ,10}$ is said to be happy if it is a bijection and for all $n \in {1,2, . . . ,10}$, $|n - f (n)| \le 1$. Compute the number of happy functions. [b]p30.[/b] Let $\vartriangle LMN$ have side lengths $LM = 15$, $MN = 14$, and $NL = 13$. Let the angle bisector of $\angle MLN$ meet the circumcircle of $\vartriangle LMN$ at a point $T \ne L$. Determine the area of $\vartriangle LMT$ . [u]Round 11[/u] [b]p31.[/b] Find the value of $$\sum_{d|2200} \tau (d),$$ where $\tau (n)$ denotes the number of divisors of $n$, and where $a|b$ means that $\frac{b}{a}$ is a positive integer. [b]p32.[/b] Let complex numbers $\omega_1,\omega_2, ...,\omega_{2019}$ be the solutions to the equation $x^{2019}-1 = 0$. Evaluate $$\sum^{2019}_{i=1} \frac{1}{1+ \omega_i}.$$ [b]p33.[/b] Let $M$ be a nonnegative real number such that $x^{x^{x^{...}}}$ diverges for all $x >M$, and $x^{x^{x^{...}}}$ converges for all $0 < x \le M$. Find $M$. [u]Round 12[/u] [b]p34.[/b] Estimate the number of digits in ${2019 \choose 1009}$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ [b]p35.[/b] You may submit any integer $E$ from $1$ to $30$. Out of the teams that submit this problem, your score will be $$\frac{E}{2 \, (the\,\, number\,\, of\,\, teams\,\, who\,\, chose\,\, E)}$$ [b]p36.[/b] We call a $m \times n$ domino-tiling a configuration of $2\times 1$ dominoes on an $m\times n$ cell grid such that each domino occupies exactly $2$ cells of the grid and all cells of the grid are covered. How many $8 \times 8$ domino-tilings are there? If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ for which there exists a function $g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)=\lfloor g(x+y)\rfloor$ for all real numbers $x$ and $y$. [i]Emil Vasile[/i]

Let $n$ points be given on a circle, and let $nk + 1$ chords between these points be drawn, where $2k+1 < n$. Show that it is possible to select $k+1$ of the chords so that no two of them intersect.

Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

Let $x$ and $y$ be real numbers such that \[ 2 < \frac{x - y}{x + y} < 5. \] If $\frac{x}{y}$ is an integer, what is its value?

Find all functions $f$ from the positive integers to the positive integers such that such that for all integers $x, y$ we have $$2yf(f(x^2)+x)=f(x+1)f(2xy).$$