IMAS 2016/2017 Junior Division Round 2

Which of the numbers below cannot be expressed as a sum of two prime numbers?

In $\triangle ABC$, $AB=AC$ and $\angle ACB=80^\circ$. Construct square $ACDE$ with the given side $AC$. Lines $BE$ and $AC$ intersect at point $F$, as shown in the figure. What is the measure of $\angle BFC$?

Alex and Charles were both sending parcels. The postage rates are as follows: For the first 10 kg and below, the postage price is 6 dollars per kg; for each successive kilogram after 10 kg, the postage price per kg is slightly lower than that of the first 10 kg. It is known that the weight of Alex’s parcel is 20% heavier than Charles’ parcel, and that the postage prices for Alex and Charles are 92 dollars and 80 dollars respectively. How much more is the postage price per kg of the first 10 kg than that of each succeeding kg above 10 kg?

It is given that $A=3x^2+3x$, $B=-x^2+x+5$ and $C=x^2+x-1$. $4A-(B-2(2B-3C)+2A)-2B=?$

On the bookshelf of Mar, there are Literature, Mathematics, History and Science books. If the number of Mathematics Books is 5 times that of the Literature books, and the number of Science books is 4 times that of the History books, which of the following is not a possible number for the total number of books on the bookshelf?

Fill in the $4\times4$ box so that the numbers 1, 2, 3, and 4 appear exactly once in each row and column. Referring to the figure below, what is the sum of the values of $A$ and $B$?

The lengths of two sides of a triangle are 6 cm and 13 cm respectively. It is known that the length of the third side is also an integer (in cm). What is the minimum perimeter (in cm) of this triangle?

It is given that Figure 1 shows a circle with diameter of 9 cm. Figure 2 shows an Olympic symbol which consists of five circles, each of diameter 9 cm. The distance between two of the tangents to the circles is 4 cm as shown. Find the length of the Olympic symbol (Figure 3)?

The diagram below is composed of many right angled isosceles triangles. Suppose an ant wants to travel from point A to point C, in how many ways can this be done if the ant is only allowed to move up, right or diagonally?

Among the 1000 positive integers from 1 to 1000 inclusive, find the number of positive integers $n$ such that $n^3+n^2+n$ is divisible by 8.

Given that $a^2+b^2+c^2=(a+b+c)^2$, where $a$, $b$ and $c$ are non-zero real numbers. What is the value of $\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}$?

Refer to the diagram below, in trapezium $ABCD$, $AD\parallel BC$. The line passing through $B$ and parallel to $CD$ intersects the line passing through $C$ and parallel to $AB$ at point $E$. Point $F$ lies inside $ABCD$ such that $\angle FAD=\angle ABC$ and $\angle FDA=\angle DCB$. Given that the area of $ABEF$ is 20 cm² and the area of $DCEF$ is 16 cm², what is the area of $ABCD$?

A 4-digit number is said to be ‘good’ if it uses exactly 3 different digits from the set {2, 0, 1, 7} (at most one of the digits used can be repeated). For example, 8712 and 7200 are said to be ‘good’ numbers, while 2017 and 7175 are not. How many ‘good’ numbers are there?

In $\triangle ABC$, point $G$ is the midpoint of segment $BC$, $BE\perp AC$, $CF\perp AB$ and lines $BE$ and $CF$ intersects at point $H$. If $\angle EGF=90^\circ$, prove that $AH=BC$.

It is known that the equation $x^2+(x+k)^2=y^2$ has positive integers solutions $(x,y)$, where $x$ and $y$ are relatively prime. If $k$ is a positive integer greater than 1, what is the minimum value of $k$?