Question bank xIMC 2011

Positive integers $a,b,c$ satisfy $ab+bc+ca+2(a+b+c)=8045$ and $abc-a-b-c=-2$. Find $a+b+c$.

A class contains students who always lie and students who never lie. Everyone knows everyone else’s type. Today each student states the type of every other student, producing the answer “liar” two hundred forty times. Yesterday the same procedure with one student absent produced “liar” two hundred sixteen times. How many students are present today?

The factors $1!\cdot2!\cdots2011!\cdot2012!$ are written on a board. Which factorial factor can be erased so the product of the remaining factors is an integer square? Here $n!$ is the product of all positive integers at most $n$.

$B,E$ lie on $AD,AC$ of triangle $ACD$, and $BC,DE$ intersect at $F$. Triangles $ABC,AED$ are congruent, with $AB=AE=1$ cm, $AC=AD=3$ cm. Find the area ratio $[ABFE]:[ADC]$.

A positive integer $n$ has exactly four positive divisors, including one and itself. Moreover $n+1$ is four times the sum of its other two divisors. Find $n$.

Jo tells Kate that three positive integers have product thirty-six and also tells their sum, yet Kate cannot determine the three numbers. What is their sum?

Circles $A,B$ of radius one centimetre touch externally. Equal-radius circles $P,Q,R,S$ have the shown external tangencies: $P$ touches $A,B,Q,S$; $Q$ touches $P,B,R$; $R$ touches $A,B,Q,S$; $S$ touches $P,A,R$. Find their common radius in centimetres.

Find the smallest positive multiple of both seven and eight whose digits are all seven or eight, with at least one of each.

A triangle has side lengths fifty, one hundred twenty and one hundred thirty centimetres. Find the area, inside and outside the triangle, of all points at distance two centimetres from at least one point on its sides. Use $\pi=22/7$.

Count the eight-digit positive integers using only zero and one, beginning with one, whose even-position digit sum equals the odd-position digit sum.

A checker on an infinite grid of unit squares first moves one square north. Odd-numbered moves are north or south; even-numbered moves east or west. On move $n$, it moves $n$ squares in one direction. After twelve moves, what is the minimum distance between the centres of its initial and final squares, in centimetres?

Real numbers $a,b,c$ satisfy $\frac{a(b-c)}{b(c-a)}=\frac{b(c-a)}{c(b-a)}=k\gt 0$. Find $\lfloor k\rfloor$.

The diagonals $AC,BD$ of quadrilateral $ABCD$ intersect at $E$. If $AE=CE$ and $\angle ABC=\angle ADC$, must $ABCD$ be a parallelogram?

For $a=1,2,\ldots,2011$, the roots of $x^2-2x-a^2-a=0$ are $\alpha_a,\beta_a$. Evaluate $\sum_{a=1}^{2011}(1/\alpha_a+1/\beta_a)$.

Fifteen rays originate at one point. What is the maximum number of obtuse angles they can form? The angle between any two rays is taken to be at most one hundred eighty degrees.

Find all real solutions of $x^2-x+1=(x^2+x+1)(x^2+2x+4)$.

Seventeen $1\times2$ dominoes cover a $5\times8$ board except six cells. Three vacant cells carry white circles in the diagram; the two black-circle cells are occupied. The other three vacant cells are in one vertical column. Which column is it?

Fill the vacant boxes with $1,2,3,4,5,6,7,9,10,11,12,13,14$ once each. Arrows from the box containing zero must point to the box containing one, those from one to two, and so on. The given boxes contain zero, eight and fifteen.

The diagram shows a $5\times8$ board with two marked cells and two example $3\times4$ subboards containing both. How many rectangular subboards of any size contain at least one marked cell?

Three avenues of widths fifteen, fourteen and thirteen metres meet at Red Triangle outside Moscow. Three swinging gates are hinged at junctions A, B, C. As shown, A and B close one avenue while C is pushed aside to allow traffic between the other two. Find the three gate lengths if every pair closes one avenue exactly.