Question 1
(a) Decompose $9^8+7^6+5^4+3^2+1$ into prime factors.
(b) Find two distinct prime factors of $2^{30}+3^{30}$.
(b) Find two distinct prime factors of $2^{30}+3^{30}$.
IMC 1999 — Keystage 3 (IWYMIC) — team
Question 2
The cards in a deck are numbered $1,3,\ldots,2n-1$. In the $k$-th step, $1\le k\le n$, $2k-1$ cards from the top of the deck are transferred to the bottom one at a time. We want the new card on top to be $2k-1$, which is then set aside. After $n$ steps, the whole deck should be set aside in increasing order. How should the deck be stacked for this to happen if (a) $n=10$; (b) $n=30$?
Question 3
(a) Express $1$ as a sum of reciprocals of distinct integers, one of which is $5$.
(b) Express $1$ as a sum of reciprocals of distinct integers, one of which is $1999$.
(b) Express $1$ as a sum of reciprocals of distinct integers, one of which is $1999$.
Question 4
(a) Show how to dissect a square into $1999$ squares, which may have different sizes.
(b) Dissect the first two shapes shown below into ten or fewer pieces that can be reassembled to form the third shape.
(b) Dissect the first two shapes shown below into ten or fewer pieces that can be reassembled to form the third shape.
Question 5
The diagram below shows a blank $5\times5$ table. Each cell is to be filled with one of $1,2,3,4,5$, so there is exactly one of each number in each row, each column and each of the two long diagonals. The score is the sum of the numbers in the four shaded cells. What is the highest possible score?
