Question bank xIMC 2010

Let $p,q,r$ be real numbers such that $p+q+r=26$ and $1/p+1/q+1/r=31$. Find $p/q+q/r+r/p+q/p+r/q+p/r$.

At a charity dinner, each person consumed half a plate of rice, a third of a plate of vegetables and a quarter of a plate of meat. Overall, sixty-five plates of food were served. How many people were there?

How many triples of positive integers $(x,y,z)$ satisfy $xyz=3^{2010}$ and $x\le y\le z\lt x+y$?

$E$ is on side $BC$ of rectangle $ABCD$. Folding along $AE$ sends $B$ to $F$ on $CD$, as shown. If $AD=16$ cm and $BE=10$ cm, find $AE$ in cm.

What is the smallest four-digit number with exactly fourteen positive divisors, including one and itself, such that one of its prime divisors has units digit three?

$f(x)$ is a fourth-degree polynomial with $f(1)=f(2)=f(3)=0$, $f(4)=6$, $f(5)=72$. Find the remainder when $f(2010)$ is divided by ten.

A circle and two semicircles, all of radius one centimetre, touch one another inside a square as shown. Find the square’s area in square centimetres.

Primes $p\ge q$ satisfy $p^3+q^3+1=p^2q^2$. What is the maximum of $p+q$?

The sum of $n$ positive integers, not necessarily distinct, is one hundred. The sum of any seven is less than fifteen. Find the minimum of $n$.

$P$ is inside triangle $ABC$ with $\angle ABP=20^\circ$, $\angle PBC=10^\circ$, $\angle ACP=20^\circ$, $\angle PCB=30^\circ$. Find $\angle CAP$ in degrees.

A farmer puts one hundred chickens and one hundred pigs into four enclosures in two rows and two columns. There are one hundred twenty heads in the first row, three hundred legs in the second row, one hundred heads in the first column and three hundred twenty legs in the second column. How many different arrangements of animal counts are possible?

Five cages in a row carry the labels Red Wolf, Silver Lion, Brown Fox, White Cow, Gray Horse, from left to right. The actual animals are a wolf, lion, fox, cow and horse with exactly those five colours. No label matches any animal, and no animal occupies or is adjacent to a cage whose label matches its type or colour. If the horse is not in the middle cage, what colour is it?

A segment divides a square into two polygons, each with an incircle. One incircle has radius six centimetres and the other is larger. Find the difference, in centimetres, between twice the segment length and the square’s side length.

Candy bags contain six, nine or twenty pieces. Buying whole bags only, what is the largest number of pieces that cannot be obtained exactly?

$S_n$ is the sum of the first $n$ terms of a sequence $a_1,a_2,\ldots$, with $a_1=2010$. If $S_n=n^2a_n$ for every positive integer $n$, find $a_{2010}$.

Solve for real numbers $w,x,y,z$: $$\begin{cases}w+8x+3y+5z=20,\\4w+7x+2y+3z=-20,\\6w+3x+8y+7z=20,\\7w+2x+7y+3z=-20.\end{cases}$$

In convex quadrilateral $ABCD$, $AB$ is the shortest side and $CD$ the longest. Prove $\angle A\gt \angle C$ and $\angle B\gt \angle D$.

Integers $m\ge n$ satisfy $m^3+n^3+1=4mn$. Find the maximum of $m-n$.

Sixty-four dots form an $8\times8$ array with adjacent same-row or same-column dots one centimetre apart. Count all rectangles of area twelve square centimetres whose vertices are among the dots.

Find the largest positive integer $n$ for which exactly one positive integer $k$ satisfies $\frac8{15}\lt \frac{n}{n+k}\lt \frac7{13}$.