2018 BIMC keystage3-IWYMIC team

How many positive integers less than 2018 can be expressed as a sum of exactly three different positive divisors of themselves?

Answer: ______ positive integers

How many pairs of positive integer $(x,y)$ are there such that $x\lt y$ and $\dfrac{x^2+y^2}{x+y}$ is an integer which is a divisor of 2835?

Show your full solution.

Answer: ______ pairs

Let $ABCD$ be a parallelogram of area $240$ $\mathrm{cm}^2$. $E$ is the midpoint of $AD$ and $H$ is the midpoint of $AB$. $G$ is a point on $BC$ such that $BG=2GC$ and $F$ is a point on $CD$ such that $DF=3FC$. $K$ is a point on $AC$ such that the area of triangle $EKF$ is $33$ $\mathrm{cm}^2$. Find the area, in $\mathrm{cm}^2$, of triangle $HKG$.



Answer: ______ $\mathrm{cm}^2$

Find the largest positive integer $m$ such that $m^4+16m+8$ is the product of two or more consecutive integers.

Show your full solution.

Answer: ______

For which positive integer value of $k$ does $\dfrac{20^k+18^k}{k!}$ attain its maximum value?

Answer: ______

The Roman army has 2018 units guarding their provinces. The Emperor was worried that when there are at least 64 units in a province, they might get together and overthrow the Emperor. So on each day, he visited one such potentially troublesome province and sent all its units to other provinces, no two going to the same province. Prove that after 64 days, there were no more provinces with at least 64 units.

The sum of all positive integers that are not relatively prime to 2018 and have exactly 2017 positive divisors is divided by 2019. Find the remainder of this division.

Answer: ______

In the diagram below, the triangle is dissected into four smaller triangles and three quadrilaterals. Each smaller triangle has area $1$ $\mathrm{cm}^2$. Find the area, in $\mathrm{cm}^2$, of quadrilateral $CA_0C_0A_1$.



Show your full solution.

Answer: ______ $\mathrm{cm}^2$

Dissect the first figure below, not necessarily along the grid lines, into three pieces such that the pieces obtained are identical up to rotation and reflection. Then dissect the second figure, not necessarily along the grid lines, into four pieces such that the pieces obtained are identical up to rotation and reflection.



Answer: First figure / Second figure

The original blank answer templates are retained below.

Tom writes down the number 1. For any number $n$ already written down, Tom writes down the numbers $5n$ and $5n+1$, provided that they are less than 1000. Jerry computes all sums of two different numbers Tom has written down. How many different sums does Jerry get?

Show your full solution.

Answer: ______