Question bank xIMC 2007

Let $a_n$ be the average of the multiples of $n$ between one and $101$. Which is largest among $a_2,a_3,a_4,a_5,a_6$?

Four people must cross a narrow bridge at night with one lantern. At most two cross together, using the slower person’s speed. If anyone remains behind, someone must return the lantern. Individual times are two, four, eight and sixteen minutes. Find the minimum total evacuation time.

In triangle $ABC$, $E\in AC,F\in AB$, and $BE,CF$ meet at $D$. Triangles $BDF,BCD,CDE$ have respective areas three, seven, seven square cm. Find the area of quadrilateral $AEDF$.

A regiment of forty-eight soldiers has uniforms for exactly half. A $6\times8$ rectangular formation has just enough interior positions to hide everyone without a uniform. New soldiers join; again half have uniforms and a different rectangular formation has exactly enough interior positions. How many joined?

The sum of $2008$ consecutive positive integers is a square. What is the minimum possible largest integer?

Two copies of a triangular paper have side lengths three, four and five cm. Each is folded through one acute-angle vertex so the two sides there coincide. The uncovered regions are shaded. Their total area is thirty-nine square cm. Find the area of the original triangle in square cm.

Find the largest positive integer $n$ such that $2^n$ divides $3^{1024}-1$.

Four straight fences of lengths one, four, seven and eight metres form a quadrilateral. What is its maximum possible area in square metres?

In the diagram, $CE=CF=EF$, $EA=BF=2AB$, and $PA=QB=PC=QC=PD=QD=1$ cm. Determine $BD$ in cm.

Use each digit $2,3,4,5,6,7,8,9$ once in the eight boxes of $$\frac1{\Box\times\Box}+\frac\Box{\Box\times\Box}+\frac\Box{\Box\times\Box}=1.$$ What is the largest of the three fractions?

For real $x$, let $\lfloor x\rfloor$ be its integer part and $\{x\}$ its fractional part. Find the sum of all positive solutions of $25\{x\}+\lfloor x\rfloor=125$.

A positive integer $n$ is good if some perfect square has decimal digit sum $n$. For example, thirteen is good because $7^2=49$ and $4+9=13$. How many good numbers are among $1,\ldots,2007$?

A $4\times4$ table has eighteen lines: four rows, four columns, and five diagonals of length two to four in each diagonal direction. Place ten counters in distinct cells. Every line with an even number of counters scores one point. Find the greatest score.

The complete road network on five cities necessarily has a crossing. So does the nine-road network joining each of three cities to each of three towns, as shown. For the fifteen roads joining all pairs of six cities, what is the minimum number of road crossings?

A decimal prime is absolute if every permutation of its digits is prime. Examples include $2,3,5,7,11,13(31),17(71),37(73),79(97),113(131,311),199(919,991),337(373,733)$. Prove no absolute prime contains all four digits one, three, seven and nine.

Place the numbers one to nine once each in the nine regions of the Olympic symbol shown, so that the sum inside every circle is fourteen.

Fourteen paper pieces are stacked as shown. Start at $B$, move to adjacent pieces, and finish at $F$, alternating a climb to a higher piece and a descent to a lower piece. Pieces may be revisited; not every piece must be visited. List a valid route.

Label the fourteen intersection points of the seven-pointed star with one to fourteen, each once, so that the four labels on each straight line have the same sum.

A three-digit number has a square ending in the same three digits. Find the sum of all distinct three-digit numbers with this property.

Find all positive integers $m,n$ with $10(m^2+1)=n^2+1$, where $m^2+1$ is prime.