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Question bank HOMC 2012

Exam papers

30 questions

Question 1

Assume that \(a-b=-(a-b)\). Then:
  1. \(a=b\)
  2. \(a\lt b\)
  3. \(a\gt b\)
  4. It is impossible to compare a and b.

Question 2

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 \(MNDC\).
  1. \(4\text{ cm}^2\)
  2. \(5\text{ cm}^2\)
  3. \(6\text{ cm}^2\)
  4. \(7\text{ cm}^2\)
  5. None of the above.

Question 3

For any positive integer \(a\), let \([a]\) denote the smallest prime factor of \(a\). Which of the following numbers equals \([35]\)?
  1. \([10]\)
  2. \([15]\)
  3. \([45]\)
  4. \([55]\)
  5. \([75]\)

Question 4

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?
  1. \(2\text{ km/h}\)
  2. \(2.5\text{ km/h}\)
  3. \(3\text{ km/h}\)
  4. \(3.5\text{ km/h}\)
  5. \(4\text{ km/h}\)

Question 5

How many different 4-digit even integers can be formed from the elements of the set \(\{1,2,3,4,5\}\)?
  1. 4
  2. 5
  3. 8
  4. 9
  5. None of the above.

Question 6

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?

Question 7

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

Question 8

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

Question 9

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

Question 10

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

Question 11

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

Question 12

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

Question 13

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

Question 14

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 \(ABC\).

Question 15

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

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.