IMC 2005 — Keystage 3 (IWYMIC) — individual

The sum of a four-digit number and its four digits is $2005$. What is this four-digit number?

In triangle $ABC$, $AB=10$ cm and $AC=18$ cm. $M$ is the midpoint of $BC$ and the line through $M$ parallel to the bisector of $\angle CAB$ cuts $AC$ at $D$. Find the length, in cm, of $AD$.

Let $x,y,z$ be positive numbers such that $x+y+xy=8$, $y+z+yz=15$ and $z+x+zx=35$. Find $x+y+z+xy$.

The total number of mushrooms gathered by eleven boys and $n$ girls is $n^2+9n-2$, with each gathering exactly the same number. Determine the positive integer $n$.

The positive integer $x$ is such that both $x$ and $x+99$ are squares of integers. Find the total value of all such integers $x$.

The lengths, in cm, of all sides of a right triangle are positive integers, and one leg is at most $20$ cm. The ratio of its circumradius to its inradius is $5:2$. Determine the maximum perimeter, in cm.

Let the larger root of $(2004x)^2-2003\cdot2005x-1=0$ be $\alpha$, and the smaller root of $x^2+2003x-2004=0$ be $\beta$. Determine $\alpha-\beta$.

Let $a\gt 0$ satisfy $a^2+1/a^2=5$. Determine $a^3+1/a^3$.

$ABCD$ is a rectangle with $AB=5$ cm. The semicircle with diameter $AB$ cuts $CD$ at two points. One of these points is $4$ cm from $A$. Find the area of $ABCD$, in square centimetres.

Let $a=9\left(n(10/9)^n-1-10/9-(10/9)^2-\cdots-(10/9)^{n-1}\right)$, where $n$ is a positive integer. If $a$ is an integer, determine its maximum value.

In a two-digit number the tens digit is greater than the units digit. Their product is divisible by their sum. What is this number?

$PQRS$ is a rectangle of area $10$ square centimetres. $A$ lies on $RS$ and $B$ on $PS$, and triangle $QAB$ has area $4$ square centimetres. Determine the smallest possible total length, in cm, of $PB$ and $AR$.

Suppose $a+bc=b+ca=c+ab=501$ for real $a,b,c$. If $M,m$ are respectively the maximum and minimum of $a+b+c$, determine $M+2m$.

The distances from a point inside a quadrilateral to its four vertices are $1,2,3,4$ cm. Determine the maximum area, in square centimetres, of such a quadrilateral.

An open-ended table has two rows. Initially $1,2,\ldots,2005$ occupy the first $2005$ squares of the first row. Each move adds the first two numbers, appends their sum to the end of that row, and drops the two used numbers into corresponding squares in the second row. Continue until one number remains in the first row, then drop it too. Determine the sum of all numbers in the second row. For example, starting with $1,2,3,4,5$ gives $1,2,3,4,5,3,7,8,15$ in the second row, whose sum is $48$.