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[Solutions] Hanoi Open Mathematics Competition 2017

Junior

  1. Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$. What is the sum $|x_1| + |x_2| + |x_3|$?.
  2. How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$?
  3. Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there?
  4. Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. What is the last digit of $S$?
  5. Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. What is the largest value of $a + b + c$?.
  6. Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime. 
  7. Determine two last digits of number $Q = 2^{2017} + 2017^2$ 
  8. Determine all real solutions $x, y, z$ of the following system of equations $$\begin{cases} x^3 - 3x &= 4 - y \\ 2y^3 - 6y& = 6 - z \\ 3z^3 - 9z &= 8 - x\end{cases}$$
  9. Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$. 
  10. Find all non-negative integers $a, b, c$ such that the roots of equations $$\begin{cases}x^2 - 2ax + b &  0 \\ x^2- 2bx + c &= 0 \\ x^2 - 2cx + a &= 0 \end{cases}$$ are non-negative integers. 
  11. Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares. 
  12. Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes? 
  13. Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$. Determine the smallest value of $$M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}.$$
  14. Given trapezoid $ABCD$ with bases $AB \parallel CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$, i.e. $CE \perp BD$ and $DE \perp AC$. By analogy, $AF \perp BD$ and $BF \perp AC$  Are three points $E$, $O$, $F$ located on the same line? 
  15. Show that an arbitrary quadrilateral can be divided into nine isosceles triangles.

Senior

  1. Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. What is the sum $|x_1| + |x_2| + |x_3|$?.
  2. How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$?
  3. The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is?.
  4. Let $a,b,c$ be three distinct positive numbers. Consider the quadratic polynomial $$P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1.$$ The value of $P (2017)$ is?.
  5. Write $2017$ following numbers on the blackboard $$-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}.$$ One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is?.
  6. Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ $$\begin{cases}x + y &= a - 1 \\ x(y + 1) - z^2 &= b \end{cases}$$
  7. Let two positive integers $x, y$ satisfy the condition $44 \mid ( x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$. 
  8. Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$. Determine the smallest value of $$M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}.$$
  9. Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting $2017$ times we obtain $2018$ pieces. We write number $2$ in every triangle, number 1 in every quadrilateral, and $0$ in the polygons. Is the sum of all inserted numbers always greater than $2017$? 
  10. Consider all words constituted by eight letters from $\{C ,H,M, O\}$. We arrange the words in an alphabet sequence. Precisely, the first word is $CCCCCCCC$, the second one is $CCCCCCCH$, the third is $CCCCCCCM$, the fourth one is $CCCCCCCO, ...,$ and the last word is $OOOOOOOO$.
    a) Determine the $2017$th word of the sequence?
    b) What is the position of the word $HOMCHOMC$ in the sequence? 
  11. Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ? 
  12. Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ? 
  13. Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $? 
  14. Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$, where $m, n \in N$.
    a) Is $P$ divisible by $24$?
    b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$? 
  15. Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$.

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MOlympiad: [Solutions] Hanoi Open Mathematics Competition 2017
[Solutions] Hanoi Open Mathematics Competition 2017
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