IMC 2004 — Keystage 3 (IWYMIC) — individual

Two circles with centres $O_1,O_2$ intersect at $A,B$. Segments $AP_1,AP_2$ are diameters of the two circles. At most how many of these six points can lie on one circle?

Let real $a,b,c$ satisfy $a^2+b^2+c^2=1$ and $a^3+b^3+c^3=1$. Determine the maximum of $a+b+c$.

In triangle $ABC$, $AB=30$ cm, $AC=32$ cm. Points $D,E,F,G$ lie respectively on $AB,AC,AD,AE$, so that $BCD,CDE,DEF,EFG,FGA$ have equal areas. Find $FD$ in centimetres.

License plates use seven distinct digits and cannot start with zero. Available molds are $0,1,2,3,5,7,9$; the mold for nine may be inverted to make six, so a plate cannot contain both six and nine. The smallest numbers are $1023567,1023576,1023579,\ldots$. Find the seven-thousandth smallest plate number.

How many positive integer pairs $(m,n)$ satisfy $m^2+n^2-16n=2004$?

You have sufficiently many $2\times5$ and $1\times3$ rectangles. How many integer pairs $(m,n)$, with $2\le n\le m$, give an $m\times n$ rectangle that cannot be tiled without overlap?

Use each integer from one to nine once in the $3\times3$ table, so the sum in each set of four cells containing an identical letter is the same. Find the central cell’s number.

Five people divide $83$ diamonds, all receiving different shares. All ten pairwise differences of the shares are distinct. Any three people together receive more than half the diamonds. How many diamonds are in the smallest share?

Sixteen dots form a $4\times4$ array. Select as many as possible so that no three selected points are the vertices of an isosceles triangle. Find the largest number selected.

A teacher chooses integers $m,n$ with $2\le m\le n$. She tells Peter only $mn$ and Steven only $m+n$. They have the following conversation:

Peter: I cannot deduce the sum.

Steven: Knowing that, I still cannot deduce the product.

Peter: Knowing that, I still cannot deduce the sum.

Steven: Now I can deduce the product. Assuming both are perfect logicians, find $n-m$.

Write $\{y\}$ for the fractional part of a real number $y$. How many real $x$ satisfy $\{(x+1)^3\}=x^3$?

For real $x,y,z$, determine the minimum of $x^2+y^2+5z^2-xy-3yz-xz+3x-4y+7z$.

In an increasing sequence of positive integers, terms in odd positions are odd and terms in even positions are even. There are seven such sequences with no term exceeding four: $(1),(3),(1,2),(1,4),(3,4),(1,2,3),(1,2,3,4)$. How many are there with no term exceeding twenty?

A triangle has sides $9,12,15$ cm. Each side is divided into $n$ equal parts, $n\ge2$. Sum the squared distances from all interior partition points to their opposite vertices. If the sum in cm$^2$ is a positive integer, find all possible $n$.

In triangle $ABC$, $BC=a$ cm, $CA=b$ cm, $AB=c$ cm. Points $D,E,F$ lie on $BC,CA,AB$ respectively, with $DE\perp CA$ and $DF\perp AB$. Let $P=BE\cap CF$. If $AP\perp BC$, determine $BD$ in terms of $a,b,c$.