IMC 2006 — Keystage 3 (IWYMIC) — individual

Colleen wants $(a+b)/c$, for positive integers $a,b,c$. Pressing $a,+,b,/,c,=$ yields eleven, whereas $b,+,a,/,c,=$ yields fourteen. She realizes division is performed before addition. Entering parentheses correctly gives the desired result. What is it?

Segment $AB$ is five cm long. In its plane, how many straight lines are two cm from $A$ and three cm from $B$?

In triangle $ABC$, $D$ is on the extension of $BC$, and $F$ is on the extension of $AB$. The bisector of $\angle ACD$ meets the extension of $BA$ at $E$; the bisector of $\angle FBC$ meets the extension of $AC$ at $G$, as shown. If $CE=BC=BG$, find $\angle ABC$ in degrees.

A teacher gives $a+b-ab=1$, saying $a$ is not an integer. Alex says $b$ is not an integer either; Brian says $b$ must be a positive integer; Colin says $b$ must be a negative integer. Who is right?

$ABCD$ is a parallelogram and $P$ is inside triangle $BAD$. If triangles $PAB,PCB$ have areas two and five square cm, find the area of triangle $PBD$ in square cm.

Suppose $x/a=y/b=z/c$, where $a,b,c,x,y,z$ are nonzero. Find $$\frac{xyz(b+c)(c+a)(a+b)}{abc(y+z)(z+x)(x+y)}.$$

A car travels at $63$ km/h on level ground, $56$ km/h uphill and $72$ km/h downhill. Going from $A$ to $B$ takes four hours, and returning takes four hours forty minutes. How far is $A$ from $B$ in km?

Square $ABCD$ has side two cm. $E,F$ are midpoints of $AB,AD$, respectively, and $G\in CF$ satisfies $3CG=2GF$. Find the area of triangle $BEG$ in square cm.

For real $x,y$, $(2x+1)^2+y^2+(y-2x)^2=1/3$. Find $x+y$.

A foreman tears one sheet into five pieces and gives them to employee one. Employee $n$ takes $n$ received pieces, tears each into five, and passes all pieces to employee $n+1$. Find $n$ such that employee $n$ receives fewer than $2006$ pieces but passes on at least $2006$.

A convex polyhedron $P$ has thirty-six edges. At every vertex, a plane cuts off a pyramid, with these cutting planes not intersecting inside $P$. Find the number of edges of the resulting convex polyhedron $Q$.

If $\sqrt{m-174}+\sqrt{m+34}=n$ for positive integers $m,n$, find the maximum value of $n$.

Four elevators in a building each stop at three floors, not necessarily consecutive or including the main floor. Every pair of floors is served together by some elevator. What is the maximum possible number of floors?

Four two-cm by four-cm rectangles are fixed in the arrangement shown. Find the radius, in cm, of the smallest circle covering them all.

Partition the integers from one to thirty into $k$ disjoint groups such that no two distinct elements within a group sum to a square. Find the minimum $k$.