Are you ready to challenge your mathematical thinking? In this post, we present the problems from the Singapore Math Kangaroo Contest 2015 – Student along with detailed solutions. This competition is known for its engaging and thought-provoking questions that test logical reasoning and problem-solving skills. Whether you’re a student preparing for math contests or simply a math enthusiast looking for an intellectual challenge, we hope these problems and explanations will help you deepen your understanding and sharpen your skills. Enjoy learning, and may this resource be beneficial for your mathematical journey!
3 Point Problems
Problem Number 1
Andrea was born in $1997,$ her younger sister Charlotte in $2001.$ The age difference of the two sisters is therefore $\cdots$ in any case.
A. less than $4$ years
B. at least $4$ years
C. exactly $4$ years
D. more than $4$ years
E. not less than $3$ years
Precisely, Andrea and Charlotte cannot be said to have an exact age difference of 4 years, unless they were born on the same date and month. Differences in birth dates and months could make their age difference less than 4 years, exactly 4 years, or more than 4 years. However, one thing is certain: their age difference is definitely not less than 3 years.
Thus, the age difference of the two sisters is therefore not less than 3 years in any case.
(Answer E)
Problem Number 2
Find the simplest form of $(a-b)^5 + (b-a)^5.$
A. 0
B. $2(a-b)^5$
C. $2(a+b)^5$
D. $2a^5-2b^5$
E. $2a^5+2b^5$
We start with the given expression:
$$(a-b)^5 + (b-a)^5.$$Since $ (b-a) = -(a-b) $, we substitute:
$$(b-a)^5 = (-(a-b))^5 = – (a-b)^5.$$Thus, the expression simplifies as follows:
$$(a-b)^5 + (b-a)^5 = (a-b)^5 -(a-b)^5 = 0.$$Therefore, the simplest form of the given expression is $\boxed{0}.$
(Answer A)
Problem Number 3
How many solutions does the equation $2^{2x}=4^{x+1}$ have?
A. $0$
B. $1$
C. $2$
D. $3$
E. Infinitely many
Given the equation $2^{2x} = 4^{x+1}.$ Since $4 = 2^2,$ we obtain
$$\begin{aligned} 2^{2x} & = (2^2)^{x+1} \\ 2^{2x} & = 2^{2x+2} \\ & = 2x & = 2x + 2 \\ 0 & = 2. \end{aligned}$$Simplifying this equation implies a false statement, namely $0 = 2.$ Therefore, it can be concluded that the equation has no solution.
(Answer A)
Problem Number 4
Diana drew a bar chart representing the quantity of the four tree species registered during a biology excursion. Jasper thinks that a circular chart would better represent the ratios of the different tree species.
What does the respective circular chart look like?
The circular chart provided in option A is the diagram that best corresponds to the given bar chart. We can determine this by matching the sector areas of the four colors in the circular chart with the heights of the bars in the bar chart.
(Answer A)
Problem Number 5
We add the $31$ integers from $2001$ to $2031$ and divide the sum by $31.$ What result do we get?
A. $2012$
B. $2013$
C. $2015$
D. $2016$
E. $2496$
By applying distributive property of integers and the concept of sum of an arithmetics series, we have
$$\begin{aligned} \dfrac{2001+2002+\cdots+2031}{31} & = \dfrac{(2000+1)+(2000+2)+\cdots+(2000+31)}{31} \\ & = \dfrac{31 \times 2000 + (1+2+\cdots+31)}{31} \\ & = \dfrac{31 \times 2000 + \frac{31}{2}(1+31)}{31} \\ & = 2000 + \dfrac12(32) \\ & = 2016. \end{aligned}$$Thus, the result we get is $\boxed{2016}.$
(Answer D)
Problem Number 6
How many of the following figures can be drawn with one continuous line without retracing any segment?
A. $0$
B. $1$
C. $2$
D. $3$
E. $4$
We can draw figures A, C, and D by tracing a continuous line without lifting the pen, following the numbered sequence as marked on each figure in order.
However, figure B cannot be drawn using a single continuous line without retracing a segment. Therefore, there are 3 figures that can be drawn with one continuous line without retracing any segment.
(Answer D)
Problem Number 7
A square piece of paper is folded along the dashed lines one after the other in any order or direction. From the resulting square one corner is cut off. Now the paper is unfolded. How many holes are in the paper?
A. $0$
B. $1$
C. $2$
D. $4$
E. $9$
Problem Number 8
A drinking glass has the shape of a truncated cone. The outside of the glass (without the base) should now be covered with colored paper.
What shape does the paper need to be in order to completely cover the whole glass without overlaps?
Problem Number 9
Three semicircles have diameters which are the sides of a right-angle triangle. Their areas are $X$ cm², $Y$ cm² and $Z$ cm², as shown. Which of the following is necessarily true?
A. $X+Y<Z$
B. $\sqrt{X}+\sqrt{Y}=\sqrt{Z}$
C. $X+Y=Z$
D. $X^2+Y^2=Z^2$
E. $X^2+Y^2=Z$
Problem Number 10
Which of the following is the complete list of the number of acute angles a convex quadrilateral can have?
A. $0, 1, 2$
B. $0, 1, 2, 3$
C. $0, 1, 2, 3, 4$
D. $0, 1, 3$
E. $1, 2, 3$
4 Point Problems
Problem Number 11
Find the value of $$\sqrt{(2015+2015)+(2015-2015)+(2015\cdot2015)+(2015:2015)}=$$
A. $\sqrt{2015}$
B. $2015$
C. $2016$
D. $2017$
E. $4030$
Problem Number 12
The $X$-axis and the graphs of the functions $f(x)=2x-2$ and $g(x)=x^2-1$ split the Cartesian plane into $\cdots \cdot$
A. $7$ regions
B. $8$ regions
C. $9$ regions
D. $10$ regions
E. $11$ regions
Problem Number 13
Ella wants to write a number in each circle in the picture such that each number is the sum of its two neighbours. Which number must Ella write in the circle with the question mark?
A. $-16$
B. $-8$
C. $-5$
D. $-3$
E. This is impossible
Problem Number 14
Given five different positive integers $a$, $b$, $c$, $d$, $e$, we know that $c:e=b$, $a+b=d$ and $ed-a$. Which of the numbers $a$, $b$, $c$, $d$, $e$ is the largest?
A. $a$
B. $b$
C. $c$
D. $d$
E. $e$
Problem Number 15
The geometric mean of a set of $n$ positive numbers is defined as the $n$-th root of the product of those numbers. The geometric mean of a set of three numbers is $3$ and the geometric mean of another set of three numbers is $12$. What is the geometric mean of the combined set of six numbers?
A. $4$
B. $6$
C. $\frac{15}{2}$
D. $\frac{15}{6}$
E. $36$
Problem Number 16
In the figure shown there are three concentric circles and two perpendicular diameters. If the three shaded figures have equal area and the radius of the small circle is $1$, what is the product of the three radii?
A. $\sqrt{6}$
B. $3$
C. $\dfrac{3\sqrt{3}}{2}$
D. $2\sqrt{2}$
E. $6$
Problem Number 17
An automobile dealer bought two cars. He sold the first one for $40\%$ more than he paid for it and the second one for $60\%$ more than he paid for it. The money he received for the two cars was $54\%$ more than what he paid for both. The ratio of the prices the dealer paid for the first and the second car was:
A. $10 : 13$
B. $20 : 27$
C. $3 : 7$
D. $7 : 12$
E. $2 : 3$
Problem Number 18
Bibi has a die with the numbers $1$, $2$, $3$, $4$, $5$ and $6$ on its six faces. Tina has a die which is special: it has the numbers $2$, $2$, $2$, $5$, $5$ and $5$ on its six faces. When Bibi and Tina roll their dice the one with the larger number wins. If the two numbers are equal it is a draw. What is the probability that Tina wins?
A. $\frac{1}{3}$
B. $\frac{7}{18}$
C. $\frac{5}{12}$
D. $\frac{1}{2}$
E. $\frac{11}{18}$
Problem Number 19
There are $2015$ marbles in a cane. The marbles are numbered from $1$ to $2015$. Marbles with equal digit sums have the same color and marbles with different digit sums have different colors. How many different colors of marbles are there in the cane?
A. $10$
B. $27$
C. $28$
D. $29$
E. $2015$
Problem Number 20
For standard dice the sum of the numbers on opposite faces is $7$. There are two identical standard dice shown in the figure. What number may be on the (not visible) face on the right (marked by the “?” sign)?
A. Only $5$
B. Only $2$
C. Either $2$ or $5$
D. Either $1$, $2$, $3$ or $5$
E. Either $2$, $3$ or $5$
5 Point Problems
Problem Number 21
The following is the multiplication table of the numbers $1$ to $10$.
$$\begin{array}{c|ccccc} x & 1 & 2 & 3 & \cdots & 10 \\ \hline 1 & 1 & 2 & 3 & \cdots & 10 \\ 2 & 2 & 4 & 6 & \cdots & 20 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 10 & 10 & 20 & 30 & \cdots & 100 \\ \end{array}$$What is the sum of all $100$ products in the complete table?
A. $1000$
B. $2025$
C. $2500$
D. $3025$
E. $5500$
Problem Number 22
The curve in the figure is described by the equation $$(x^2+y^2-2x)^2=2(x^2+y^2).$$Which of the lines $a$, $b$, $c$, $d$ represents the y-axis?
A. $a$
B. $b$
C. $c$
D. $d$
E. none of these
Problem Number 23
When reading the following statements from the left to the right, what is the first statement which is true?
A. (C) is true
B. (A) is true
C. (E) is false
D. (B) is false
E. $1 + 1 = 2$
Problem Number 24
How many regular polygons exist such that their angles (in degrees) are integers?
A. $17$
B. $18$
C. $22$
D. $25$
E. $60$
Problem Number 25
How many $3$-digit positive integers can be represented as the sum of exactly nine different powers of $2$?
A. $1$
B. $2$
C. $3$
D. $4$
E. $5$
Problem Number 26
How many triangles ABC with $\angle ABC=90°$ and $AB=20$ exist such that all sides have integer lengths?
A. $1$
B. $2$
C. $3$
D. $4$
E. $6$
Problem Number 27
In the rectangle ABCD shown in the figure, $M_1$ is the midpoint of CD, $M_2$ is the midpoint of $AM_1$, $M_3$ is the midpoint of $BM_2$ and $M_4$ is the midpoint of $CM_3$. Find the ratio between the areas of the quadrilateral $M_1M_2M_3M_4$ and of the rectangle ABCD.
A. $\frac{7}{16}$
B. $\frac{3}{16}$
C. $\frac{7}{32}$
D. $\frac{9}{32}$
E. $\frac{1}{5}$
Problem Number 28
Blue and red rectangles are drawn on a blackboard. Exactly $7$ of the rectangles are squares. There are $3$ red rectangles more than blue squares. There are $2$ red squares more than blue rectangles. How many blue rectangles are there on the blackboard?
A. $1$
B. $3$
C. $5$
D. $6$
E. $10$
Problem Number 29
$96$ members of a counting club are standing in a large circle. They start saying numbers $1$, $2$, $3$, etc. in turn, going around the circle. Every member that says an even number steps out of the circle and the rest continue, starting the second round with $97$. They continue in this way until only one member is left. Which number did this member say in the first round?
A. $1$
B. $17$
C. $33$
D. $65$
E. $95$
Problem Number 30
In the word KANGAROO Bill and Bob replace the letters by digits, so that the resulting numbers are multiples of $11$. They each replace different letters by different digits and the same letters by the same digits ($K≠0$). Bill obtains the largest possible such number and Bob the smallest. In both cases one of the letters is replaced by the same digit. Which digit is this?
A. $0$
B. $3$
C. $4$
D. $5$
E. $6$