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IMAS 2012–2013 — Junior Division — Round 2

Question 1

In a promotional sale, anyone who buys a cup of juice at the regular price of 7 dollars can get a second cup of juice by paying 1 more dollar. What is the minimum number of dollars a party of 9 people must pay if each of them wants a cup of juice?
  1. 32
  2. 36
  3. 39
  4. 40
  5. 63

Question 2

Into the expression $y=\frac{x^2}{1+x^2}$, we substitute for $x$ the numbers $\frac1{2012},\frac1{2008},\frac1{2004},\ldots,\frac14,4,8,12,\ldots,2012$. What is the sum of all the $y$ values?
  1. 1
  2. 16
  3. 251.5
  4. 503
  5. 2012

Question 3

In a flower shop, each Carnation sells for 3 dollars and each Rose sells for 4 dollars. A bouquet is a combination of these two kinds of flowers. How many different bouquets selling for 60 dollars are there?
  1. 4
  2. 5
  3. 6
  4. 7
  5. 8

Question 4

A rectangular strip, 30 cm in length and 3 cm wide, is folded in a pattern shown in diagram (2), producing a right angle $\angle ACB$. After the strip is completely folded as shown in diagrams (3) and (4), the lengths of $AM$ and $GM$ are equal. What is the length of $AC$ in diagram (1)?

Question illustration
  1. 7.5
  2. 10.5
  3. 13.5
  4. 14.5
  5. 16.5

Question 5

Let $a,b,c$ be rational numbers such that $c=-\frac{ab}{a+b}$. Which of the following expressions is correct?
  1. $a+b+c=a^3+b^3+c^3$
  2. $(a+b+c)^2=a^2+b^2+c^2$
  3. $(a+b+2c)^2=a^2+b^2-4c^2$
  4. $(a+b+c)^3=a^3+b^3+c^3$
  5. $(a+b+c)(a^2+b^2+c^2)=a^3+b^3+c^3+abc$

Question 6

In an isosceles triangle $ABC$, $AB=AC$. Point $D$ lies on side $AC$, so that $AD=DB=BC$. What is the measure of $\angle BAC$? Answer in degrees.

Question 7

The diagram below shows three overlapping regular hexagons, each with side length 12 cm$^2$. One of the vertices of the middle hexagon coincides with the center $O_1$ of the hexagon on the left, while one of the vertices of the hexagon on the right coincides with the center $O_2$ of the middle hexagon. What is the total area of the two shaded parts? Answer in cm$^2$.

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Question 8

Let $a$ and $b$ be real numbers such that $3^a=2013$ and $671^b=2013$. What is the value of $\frac1a+\frac1b$?

Question 9

In the diagram below, $ABCD$ is a right-angled trapezoid such that $AD\parallel BC$ and $E$ is the midpoint of $CD$. If $BE=20$ cm and $AB=AD+BC$, determine the area of trapezoid $ABCD$, in cm$^2$?

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Question 10

If a 7-digit number $\overline{20ab13c}$ is divisible by 792, what is the value of $c(a+b)$?

Question 11

In a party each of the participant handshakes with four women and six men. Given that the known number of handshakes between opposite-sex participants is seven times less than the number of handshakes between same-sex participants, how many men are there in the party?

Question 12

We put 130 identical balls into several identical boxes, such that the number of balls in each box is at least 10 but at most 20. The numbers of balls in the boxes are all different. How many distinct ways of putting these balls in the boxes are there?

Question 13

Let $a$ and $b$ be non-negative integers less than 100. If $a-2b$ is a positive prime number and $2ab$ is a perfect square number, what is maximum value of $a+b$?

Question 14

Let $k$ be a real number. The product of all the real roots of the equation $x^4+2x^3+(3+k)x^2+(2+k)x+2k=0$ is $-2012$. Find the sum of the squares of the real roots. Detailed solutions are needed.

Question 15

A $9\times11$ chessboard may be covered without overlap with a combination of the following three shapes. What is the minimum number of copies of the piece consisting of three squares must be used? Detailed solutions are needed.

Question illustration

Question illustration