It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?

The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: $\bullet$ No divisor of a previously written number may be written. $\bullet$ The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$. $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale).

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

How many $3 \times 5$ rectangular pieces of cardboard can be cut from a $17 \times 22$ rectangular piece of cardboard, when the amount of waste is minimised?

In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.

In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks. How many cows will eat all the grass on six hectares of land in six weeks, together with all the grass that regrows there over the six weeks? (Assume: $\bullet$ the quantity of grass on each hectare is the same when the cows begin to graze, $\bullet$ the rate of growth of the grass is uniform during the time of grazing, $\bullet$ the cows eat the same amount of grass each week.)

Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.

In a city at every square exactly three roads meet, one is called street, one is an avenue, and one is a crescent. Most roads connect squares but three roads go outside of the city. Prove that among the roads going out of the city one is a street, one is an avenue and one is a crescent.