IMAS 2014–2015 — Junior — Round 2

Which of the following expressions is equal to $(a+1)(b-1)+(b+1)(c-1)+(c+1)(a-1)$?

In triangle $ABC$, $D$ is the midpoint of $BC$. $E$ is an arbitrary point on $CA$, and $F$ is the midpoint of $BE$. If the area of triangle $ABC$ is 120 cm² and the area of the quadrilateral $AFDC$ is 80 cm², what is the area, in cm², of triangle $BDF$?

Let $m$ be a positive integer such that $m^3$ can be expressed as a sum of $m$ consecutive odd integers. For instance, $2^3=3+5$, $3^3=7+9+11$ and $4^3=13+15+17+19$. If 999 is one of the consecutive odd integers in the expression for $m^3$, what is the value of $m$?

The perimeter of an equilateral triangle is $a$ cm while the perimeter of a square is $b$ cm. If the area of the square is half the area of the triangle, what is the value of $\frac{a^2}{b^2}$?

Mindy has two boxes, containing 0 and $n$ pieces of candy respectively, where $n$ is a positive integer. She adds 4, 3 and 2 pieces of candy to one of the boxes in that order, always adding to the box containing fewer pieces of candies. If the two boxes have the same number of pieces of candy, then she adds to either of them. In the end, there is 1 more piece of candy in one box than in the other. How many possible values of $n$ are there?

Let $a$, $b$ and $c$ be real numbers such that $x^2+5x-3$ is one of the factors of the polynomial $x^3+ax^2+bx+c$. What is the numerical value of $a+b+2c$?

How many triples of integers $(x,y,z)$ are such that $|xyz|=6$?

In triangle $ABC$, $AB=7$ cm, $AC=8$ cm and $BC=9$ cm. A circle with centre $A$ intersects $AB$ at $F$ and $AC$ at $E$. The circles with centres $B$ and $C$ and radii $BF$ and $CE$, respectively, are tangent to each other at a point $D$ on $BC$. What is the total area, in cm², of these three circles? (Taking $\pi=3.14$)

There are 2 counters in the first row, and each subsequent row has one more counter than the preceding row. If there are 2015 counters altogether, how many rows of counters are there?

In a book fair, the organizers give a book to each participant. Each male participant gives every other male participant a book, and each female participant gives every other female participant a book. If the total number of books received by the male participants is 31 more than the total number of books received by the female participants, how many participants are there altogether?

$D$ is a point on the side $BC$ of triangle $ABC$ such that $\angle BAD=76^\circ$. When the point $C$ is reflected across $AD$ to the point $C'$, $ABC'D$ is a parallelogram. What is the measure, in degrees, of $\angle ADC$?

Let $k$ be a non-zero integer such that the equation $x+\frac{9k^2-81}{x}=10k$ has two distinct integer roots. What is the difference when the smaller root is subtracted from the larger one?

The integers 1, 2, 3, …, 20 are divided into two groups. The sum of all the numbers in one group is $n$, while the product of all the numbers in the other group is also $n$. What is the maximum value of $n$?

$E$ is a point on the side $BC$ and $F$ is a point on the side $CD$ of a square $ABCD$ such that the perimeter of triangle $CEF$ is equal to half the perimeter of $ABCD$. $G$ is the point on $AE$ such that $FG$ is perpendicular to $AE$, and $H$ is the point on $FG$ such that $AH=EF$. Prove that $AH$ is perpendicular to $EF$.

Each side of an equilateral triangle is divided into 5 equal parts by 4 points, and these points are joined by lines parallel to the sides of the triangle, dividing into 25 small equilateral triangles. A tetriamond is a shape formed of 4 small equilateral triangles joined edge to edge. There are three tetriamonds as shown in the diagram below on the left.



(a) Show that if 7 of the small triangles are painted, then it may be impossible to fit any tetriamond inside the large triangle without covering up any part of the painted small triangles. (4 marks)



(b) Prove that if 6 of the small triangles are painted, then it is always possible to fit a tetriamond inside the large triangle without covering up any part of the painted small triangles. (16 marks)