IMC 2001 — Keystage 3 (IWYMIC) — individual

Find all integers $n$ such that $1+2+\cdots+n$ is a three-digit number with identical digits.

$ABCDE$ is a convex pentagon with $CD=DE=4$ cm, $EA=AB=BC=2$ cm and $\angle A=\angle B=120^\circ$. Find its area in square centimetres.

A $6$ cm by $6$ cm square can cover at most $60\%$ of a triangle. When the triangle is placed on the square, it can cover at most $\frac23$ of the square. Find the triangle’s area in square centimetres.

Four consecutive positive integers have the following properties: the smallest is a multiple of $5$, the second of $7$, the third of $9$, and the largest of $11$. Find the minimum possible largest number.

Between five and six o’clock, a lady looked at her watch and mistook the hour hand for the minute hand and vice versa. She thought the time was approximately $55$ minutes earlier. Exactly how many minutes earlier was the mistaken time?

The incircle of triangle $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Its radius is $4$ cm. If the lengths $BD,CE,AF$ in cm are consecutive integers, find the perimeter in cm.

Determine the largest prime $p$ for which integers $x,y$ exist with $p+1=2x^2$ and $p^2+1=2y^2$.

Find the largest real solution of $\sqrt{3x^2-18x+52}+\sqrt{2x^2-12x+162}=\sqrt{-x^2+6x+280}$.

Simplify $\sqrt{12-\sqrt{24}+\sqrt{39}-\sqrt{104}}-\sqrt{12+\sqrt{24}+\sqrt{39}+\sqrt{104}}$ to one numerical value.

Let $m=1010101\cdots01$, with the digit $1$ occurring $k$ times. Find the least $k$ such that $1001001001001$ divides $m$.

Let $a,b$ be unequal positive real numbers, $A=(a+b)/2$, $G=\sqrt{ab}$. Prove $G\lt \frac{(a-b)^2}{8(A-G)}\lt A$.

Find the range of $p$ for which $3^{2x}-3^{x+1}=p$ has two distinct positive real roots.

A square has all four vertices on the perimeter of an acute scalene triangle: one on each of two sides and two on the third side. To make the square as large as possible, should the side containing two vertices be the longest, shortest, or neither? Justify your answer.