The age of Max now times the age of Mini a year from now is the square of an integer. The age of Max a year from now times the age of Mini now is also the square of an integer. If Mini is 8 years old now, and Max is now older than 1 but younger than 100, how old is Max now?
In a choir, more than $\frac{2}{5}$ but less than $\frac{1}{2}$ of the children are boys. What is the smallest possible number of children in this choir?
Each girl wants to ride a horse by herself, but there are only enough horses for $\frac{10}{13}$ of them. If the total number of legs of all the horses and girls is 990, how many girls will have to wait for their turns?
Clearly, $\frac{23}{30}=\frac{57}{78}$ is incorrect. However, if the same positive integer is subtracted from each of 23, 30, 57 and 78, then it will be correct. What is the number to be subtracted?
A cat has caught three times as many black mice as white mice. Each day, she eats 6 black mice and 4 white mice. After a few days, there are 60 black mice and 4 white mice left. How many mice has the cat caught?
$M$ is the midpoint of the side $CD$ of a square $ABCD$ of side length 24 cm. $P$ is a point such that $PA=PB=PM$. What is the minimum length, in cm, of $PM$?
In a party, every two people shake hands except for Bob, who only shakes hands with some of the people. No two people shake hands more than once. If the total number of handshakes is 2014, with how many people does Bob shake hands?
The cost of a ticket for a concert is 26 dollars for an adult, 18 dollars for a youth and 10 dollars for a child. The total cost of a party of 131 people is 2014 dollars. How many more children than adults are in the party?
Two overlapping squares with parallel sides are such that the part common to both squares has an area of $4\,\mathrm{cm}^2$. This is $\frac{1}{9}$ the area of the larger square and $\frac{1}{4}$ of the area of the smaller square. What is the minimum perimeter, in cm, of the eight-sided figure formed by the overlapping squares?
The number of stars in the sky is
$$8\times12+98\times102+998\times1002+\cdots+99\cdots98\times100\cdots02.$$
In the last term, there are 2014 copies of the digit 9 in $99\cdots98$ and 2014 copies of the digit 0 in $100\cdots02$. What is the sum of the digits of the number of stars?
In a triangle $ABC$, $D$ is a point on $BC$ and $F$ is a point on $AB$. The point $K$ of reflection of $B$ across $DF$ is on the opposite side of $AC$ to $B$. $AC$ intersects $FK$ at $P$ and $DK$ at $Q$. The total area of triangles $AFP$, $PKQ$ and $QDC$ is $10\,\mathrm{cm}^2$. If we add to this the area of the quadrilateral $DFPQ$, we obtain $\frac{2}{3}$ of the area of $ABC$. What is the area, in $\mathrm{cm}^2$, of triangle $ABC$?
After Nadia goes up a hill, she finds a level path on top of length 2.5 km. At the end of it, she goes down the hill to a pond. Later, she goes back along the same route. Her walking speed is 5 kph, but it decreases to 4 kph going up the hill, and increases to 6 kph going down the hill. Her outward journey takes 1 hour 36 minutes but her return journey takes 1 hour 39 minutes. She does not stop anywhere at any time. What is the length, in km, from start point to the pond?
Five colours are available for the painting of the six faces of a cube. One colour is used to paint two of the faces, while each of the other four colours is used to paint one face. How many differently painted cubes can there be? Two cubes painting the same colours on corresponding faces after rotation or flip are not considered to be different.
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.
