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[Solutions] Hanoi Open Mathematics Competition 2015

Junior

  1. What is the $7$th term of the sequence $\{-1, 4,-2, 3,-3, 2,...\}$?
  2. The last digit of number $2017^{2017} - 2013^{2015}$ is?
  3. The sum of all even positive integers less than $100$ those are not divisible by $3$ is?
  4. A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is?.
  5. Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $$a + b + c = (a - b)(b- c)(c - a) = m \pmod 27$$ then $m$ is?
  6. Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$. Prove that $$1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4.$$
  7. Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$. 
  8. Solve the equation $$(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3$$
  9. Let $a, b,c$ be positive numbers with $abc = 1$. Prove that $$a^3 + b^3 + c^3 + 2[(ab)^3 + (bc)^3 + (ca)^3] \ge 3(a^2b + b^2c + c^2a).$$
  10. A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle. 
  11. Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I$, $K$, $H$ be feet of perpendiculars from $B$, $O$, $C$ to $AD$, respectively. Prove that $$AD \times BI \times CH \le AC \times BD \times OK.$$
  12. Give a triangle $ABC$ with heights $h_a = 3 cm$, $h_b = 7 cm$ and $h_c = d cm$, where $d$ is an integer. Determine $d$. 
  13. Give rational numbers $x, y$ such that $$(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0.$$ Prove that $\sqrt{1 + xy}$ is a rational number. 
  14. Determine all pairs of integers $(x, y)$ such that $$2xy^2 + x + y + 1 = x^2 + 2y^2 + xy.$$
  15. Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$. Determine the maximum value of $$M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$$

Senior

  1. The sum of all even positive integers less than $100$ those are not divisible by $3$ is?
  2. A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is?.
  3. Suppose that $a > b > c > 1$. One of solutions of the equation $$\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$$ is?
  4. Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is?
  5. The last digit of number $2017^{2017} - 2013^{2015}$ is?
  6. Let $a, b, c \in [-1, 1] $ such that $1 + 2abc \ge a^2 + b^2 + c^2$. Prove that $$1 + 2a^2b^2c^2 \ge a^4 + b^4 + c^4.$$
  7. Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$. 
  8. Solve the equation $$(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$$
  9. Let $a, b,c$ be positive numbers with $abc = 1$. Prove that $$a^3 + b^3 + c^3 + 2[(ab)^3 + (bc)^3 + (ca)^3] \ge 3(a^2b + b^2c + c^2a).$$
  10. A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle. 
  11. Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I$, $K$, $H$ be feet of perpendiculars from $B$, $O$, $C$ to $AD$, respectively. Prove that $$AD \times BI \times CH \le AC \times BD \times OK.$$
  12. Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp BC$. Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$, $d =a^{2015} + b^{2015}$. Prove that $c$ and $d$ are relatively prime numbers. 
  13. Determine all pairs of integers $(x, y)$ such that $$2xy^2 + x + y + 1 = x^2 + 2y^2 + xy.$$
  14. Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $$M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$$

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MOlympiad: [Solutions] Hanoi Open Mathematics Competition 2015
[Solutions] Hanoi Open Mathematics Competition 2015
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