IMSO 2011 International Mathematics — Short Answer

The sum of the three smallest prime numbers and another prime number $n$ is 2021. Find the value of $n$.

The average of $x$ and $y$ is 19. The average of $a,b,c$ is 14. Find the average of $x,y,a,b,c$.

The figure is made up of semicircles of diameter 2 cm, 4 cm and 8 cm. What fraction of the figure is shaded?

The sum of the numbers $A,B,C$ is 390. Given $A$ is 3 times $B$ and $A$ is one third of $C$, find $C$.

What is the smallest number that can be expressed as the sum of two squares in two different ways?

How many rectangles are there in the $3\times3$ grid shown?

Find the smallest positive integer with exactly 30 factors.

Find the value of $$\begin{aligned}&\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac67\right)\left(\frac12+\frac{2009}{2010}+\frac{2010}{2011}+\frac25\right)\\[-1mm]&\quad-\left(\frac12+\frac{2009}{2010}+\frac{2010}{2011}+\frac67\right)\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac25\right).\end{aligned}$$

The number 119 is very amazing. When divided by 2,3,4,5,6 it leaves remainders 1,2,3,4,5 respectively. Find the smallest three-digit number, larger than 119, which has this property.

Pedro can finish a job in 14 minutes while his younger brother Juan can finish the same job in 7 minutes. How long will it take the two of them to finish the job together?

In the diagram shown, find the measure of $\angle a+\angle b+\angle c+\angle d+\angle e$.

The sum of the two digits of Emma’s age this year is 5. Seven years from now, her age will be 2 less than the reverse of the digits of her age this year. How old is Emma now?

Half of the punch in the bowl is pure mango juice. When 6 more cups of pure mango juice are added, $\frac23$ of the resulting punch is pure mango juice. How many cups of punch were in the bowl to start with?

Five test scores have a mean of 88, a median of 89, and a mode of 93. What is the sum of the two lowest test scores?

The area of a rectangle with integer sides is 2011 cm². Find its largest possible perimeter, in cm.

In the grid shown, how many paths are there from corner A to corner B that only have steps to the right or up and do not pass through neither C nor D?

Chords EF and GH are perpendicular to each other. A and B are centers of circle A and circle B respectively. Circle A and circle B are tangent to the chords, as shown. If their radii are 12 and 5 respectively, find AB. Enter an exact radical, for example 3*sqrt(2).

Six points A,B,C,D,E,F are arranged as shown. How many triangles can be formed with any 3 of the 6 points as vertices?

A square piece of paper has area 96 cm². It is folded along the dotted line MN. The shaded area in the resulting figure is one-half of the total visible area. Find MN. Enter an exact radical, for example 3*sqrt(2).

What is the maximum number of circles of radius 2 that can be packed without overlap into a circle of radius 6?