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[Solutions] Hanoi Open Mathematics Competition 2016

Junior

  1. If $2016 = 2^5 + 2^6 + ...+ 2^m$ then $m$ is equal to?.
  2. The number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$ is?
  3. Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$. Find the greatest value of $M = a^2 + b^2 - ab$.
  4. A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer.
  5. There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to?
  6. Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$. 
  7. Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points? 
  8. Find all positive integers $x,y,z$ such that $$x^3 - (x + y + z)^2 = (y + z)^3 + 34$$
  9. Let $x, y,z$ satisfy the following inequalities $$\begin{cases} | x + 2y - 3z| &\le 6 \\ | x - 2y + 3z| &\le 6 \\ | x - 2y - 3z| &\le 6 \\ | x + 2y + 3z| &\le 6 \end{cases}$$ Determine the greatest value of $M = |x| + |y| + |z|$. 
  10. Let $h_a$, $h_b$, $h_c$ and $r$ be the lengths of altitudes and radius of the inscribed circle of $\vartriangle ABC$, respectively. Prove that $$h_a + 4h_b + 9h_c > 36r.$$
  11. Let be given a triangle $ABC$ and let $I$ be the middle point of $BC$. The straight line $d$ passing $I$ intersects $AB$, $AC$ at $M$, $N$, respectively. The straight line $d'$ ($\ne d$) passing $I$ intersects $AB$, $AC$ at $Q$, $P$, respectively. Suppose $M$, $P$ are on the same side of $BC$ and $MP$, $NQ$ intersect $BC$ at $E$ and $F$, respectively. Prove that $IE = I F$. 
  12. In the trapezoid $ABCD$, $AB // CD$ and the diagonals intersect at $O$. The points $P$, $Q$ are on $AD$, $BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$. 
  13. Let $H$ be orthocenter of the triangle $ABC$. Let $d_1$, $d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB$, $AC$ at $D$, $E$ and the line $d_2$ intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$ if and only if $F$ is the midpoint of segment $BC$. 
  14. Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number. 
  15. Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015$, $f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

Senior

  1. How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$.
  2. Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table. Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 \min(a,b,c)$. After $672$ times, on the table there is only one number $m$.
  3. Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is?
  4. A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer.
  5. There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to?
  6. Let $A$ consist of $16$ elements of the set $\{1, 2, 3,..., 106\}$, so that the difference of two arbitrary elements in $A$ are different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements of $A$ for which their difference equals to $3$. 
  7. Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points? 
  8. Determine all $3$-digit numbers which are equal to cube of the sum of all its digits. 
  9. Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in \mathbb Z$. Prove that there exist two relative prime numbers $m$, $n$ such that $abc =\dfrac{m^2}{n^3}$ . 
  10. Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number. 
  11. Let $I$ be the incenter of triangle $ABC$ and $\omega$ be its circumcircle. Let the line $AI$ intersect $\omega$ at point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac12 \angle BAC$ . Let $X$ be the second point of intersection of line $EI$ with $\omega$ and $T$ be the point of intersection of segment $DX$ with line $AF$ . Prove that $TF \cdot AD = ID \cdot AT$ . 
  12. Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$). 
  13. Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$. 
  14. Let $f (x) = x^2 + px + q$, where $p, q$ are integers. Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$. 
  15. Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$. Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.

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MOlympiad: [Solutions] Hanoi Open Mathematics Competition 2016
[Solutions] Hanoi Open Mathematics Competition 2016
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