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Hanoi Open Mathematics Competition 2012 Senior

Question 1

Let \(x=\dfrac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}\). The value of \(H=(1+x^5-x^7)^{2012^{3^{11}}}\) is
  1. 1
  2. 11
  3. 21
  4. 101
  5. None of the above.

Question 2

Compare \(P=2^\alpha,Q=3,T=2^\beta\), where \(\alpha=\sqrt2\), \(\beta=1+\dfrac1{\sqrt2}\).
  1. \(P\lt Q\lt T\)
  2. \(T\lt P\lt Q\)
  3. \(P\lt T\lt Q\)
  4. \(T\lt Q\lt P\)
  5. \(Q\lt P\lt T\)

Question 3

In trapezoid \(ABCD\), the bases are \(BC=3\text{ cm}\), \(DA=6\text{ cm}\), with \(AD\parallel BC\). The segment \(EF\), where \(E\in AB,F\in CD,EF\parallel AD\), passes through \(M=AC\cap BD\). Find its length.
  1. \(3.5\text{ cm}\)
  2. \(4\text{ cm}\)
  3. \(4.5\text{ cm}\)
  4. \(5\text{ cm}\)
  5. None of the above.

Question 4

What is the largest integer less than or equal to \(4x^3-3x\), where \(x=\dfrac12\left(\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}\right)\)?
  1. 1
  2. 2
  3. 3
  4. 4
  5. None of the above.

Question 5

Let \(f(x)\) satisfy \(f(x)+2f\left(\dfrac{x+2010}{x-1}\right)=4020-x\) for every \(x\ne1\). The value of \(f(2012)\) is
  1. 2010
  2. 2011
  3. 2012
  4. 2014
  5. None of the above.

Question 6

For \(n=2,3,\ldots\), put \(A_n=\left(1-\dfrac1{1+2}\right)\left(1-\dfrac1{1+2+3}\right)\cdots\left(1-\dfrac1{1+2+\cdots+n}\right)\). Determine all integers \(n\ge2\) for which \(1/A_n\) is an integer.

Question 7

Prove that the decimal integer \(a=\underbrace{11\ldots1}_{2012\text{ digits}}\underbrace{55\ldots5}_{2011\text{ digits}}6\) is a perfect square.

Question 8

Determine the greatest real number \(m\) such that the system \(x^2+y^2=1\), \(|x^3-y^3|+|x-y|=m^3\), has a real solution.

Question 9

Let \(P\) be the common point of the three internal bisectors of triangle \(ABC\). The line through \(P\) perpendicular to \(CP\) meets \(AC,BC\) at \(M,N\), respectively. If \(AP=3\text{ cm}\), \(BP=4\text{ cm}\), compute \(AM/BN\).

Question 10

The equation \(x^3+px^2+qx+1=0\), with rational \(p,q\), has three real roots \(x_1,x_2,x_3\), where \(x_3=2+\sqrt5\). Compute \(p,q\).

Question 11

The equation \(x^3+px^2+qx+r=0\) has three real roots \(x_1,x_2,x_3\), where \(p,q,r\) are integers. Put \(S_n=x_1^n+x_2^n+x_3^n\) for \(n=1,2,\ldots\). Prove that \(S_{2012}\) is an integer.

Question 12

In an isosceles triangle \(ABC\) with base \(AB\), let \(M\in BC\). Let \(O\) be its circumcenter and \(S\) its incenter. If \(SM\parallel AC\), prove that \(OM\perp BS\).

Question 13

A cube of side \(3\text{ cm}\) is painted red and cut into \(3\times3\times3=27\) cubes of side \(1\text{ cm}\). Let \(a,b,c,d\) be the numbers of small cubes with respectively zero, one, two and three painted faces. Determine \(a-b-c+d\).

Question 15

Determine the smallest value of \(M=xy-yz-zx\), where real numbers \(x,y,z\) satisfy \(x^2+2y^2+5z^2=22\).