Exactly one pair of brackets is to be inserted into the expression
$$2\times2-2\times2-2\times2-2\times2-2\times2.$$
The left bracket must come before a 2 and the right bracket after a 2. Determine the largest possible value of the resulting expression.
The diagram below shows a right isosceles triangle partitioned into four shaded right isosceles triangles and a number of squares. The side lengths of all the squares are positive integers. The smallest ten squares are of side length 1 cm. Determine the total area, in $\mathrm{cm}^2$, of the four shaded triangles.
Each of the five families living in apartments 2, 3, 4, 6 and 12 in a building will adopt one of five cats, whose ages are 1, 2, 3, 4 and 6. The adopting family's apartment number must be divisible by the age of the cat. Find all possible adoption schemes.
The source answer template has columns for cats aged 1, 2, 3, 4 and 6 years, respectively.
The positive integers $1,2,\ldots,2014$ strung together form a very long multi-digit number $12345678910111213\ldots201220132014$. A seven-digit multiple of 11 is obtained by erasing all the digits before it and all the digits after it. Determine the smallest possible value of this seven-digit number, which may not start with the digit 0.
The diagram below shows a hexagonal configuration of 19 dots in an equilateral triangular grid.
**(a)** Determine the number of equilateral triangles of different sizes with all three vertices among the 19 dots. Draw an equilateral triangle of each size on the diagram.
**(b)** Determine the number of equilateral triangles of each size.
Each of teams $A,B,C,D$ and $E$ plays against every other team exactly once. A win is worth 3 points, a draw 1 point and a loss 0 points. At the end of the tournament, no two teams have the same number of points. $A$ has the highest number of points despite losing to $B$. Neither $B$ nor $C$ loses any game, but $C$ has fewer points than $D$. How many points does $E$ have?
$P$ is a point inside a square $ABCD$ of side length 8 cm. What is the largest possible value of the area, in $\mathrm{cm}^2$, of the smallest one among the six triangles $PAB,PBC,PCD,PDA,PAC$ and $PBD$?
A sequence of 2014 two-digit numbers is such that each is a multiple of 19 or 23, and the tens digit of any number starting from the second is equal to the units digit of the preceding number. If the last number in the sequence is 23, determine the first number in the sequence.
There are ten real coins all of the same weight. There is a fake coin which is heavier than a real coin, and another fake coin which is lighter than a real coin. You cannot tell the coins apart. Explain how, in four weighings using a balance, you may determine whether the total weight of the two fake coins is greater than, equal to or less than the total weight of two real coins?
