Hanoi Open Mathematics Competition 2012 Junior

Assume that \(a-b=-(a-b)\). Then:

Given a parallelogram \(ABCD\) of area \(12\text{ cm}^2\), the line through \(A\) and the midpoint \(M\) of \(BC\) meets \(BD\) at \(N\). Compute the area of quadrilateral \(MNDC\).

For any positive integer \(a\), let \([a]\) denote the smallest prime factor of \(a\). Which of the following numbers equals \([35]\)?

A man travels from town \(A\) to town \(E\) through \(B,C,D\), with uniform speeds \(3,2,6,3\) km/h on the horizontal, uphill, downhill and horizontal roads, respectively. If the road can be classified as horizontal, uphill, downhill and horizontal and the total length of each type of road is the same, what is his average speed?

How many different 4-digit even integers can be formed from the elements of the set \(\{1,2,3,4,5\}\)?

At 3:00 A.M. the temperature was \(13^\circ\) below zero. By noon it had risen to \(32^\circ\). What is the average hourly increase in temperature?

Find all integers \(n\) such that \(60+2n-n^2\) is a perfect square.

Given a triangle \(ABC\) and points \(K\in AB\), \(N\in BC\) such that \(BK=2AK\), \(CN=2BN\), let \(Q=AN\cap CK\). Compute \(\dfrac{[ABC]}{[BCQ]}\).

Evaluate the integer part of \(H=\sqrt{1+2011^2+\dfrac{2011^2}{2012^2}}+\dfrac{2011}{2012}\).

Solve \(\dfrac1{(x+29)^2}+\dfrac1{(x+30)^2}=\dfrac{13}{36}\).

Let \(a_1=5,a_2=8\), and \(a_{n+1}=a_n+3a_{n-1}\) for \(n=2,3,\ldots\). Calculate \(\gcd(a_{2011},a_{2012})\).

Find all positive integers \(P\) such that the sum and product of all its divisors are \(2P\) and \(P^2\), respectively.

Determine the greatest value of \(M=11xy+3xz+2012yz\), where \(x,y,z\) are nonnegative integers satisfying \(x+y+z=1000\).

In triangle \(ABC\), \(\angle A=90^\circ\), and the bisectors of \(B,C\) meet at \(I\). Let \(IH\perp BC\), with \(H\in BC\). If \(HB=5\text{ cm}\), \(HC=8\text{ cm}\), compute the area of \(ABC\).

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