2024 Fermat Contest
(Grade 11)
Wednesday, February 28, 2024
(in North America and South America)
Thursday, February 29, 2024
(outside of North American and South America)
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The value of is
If , the value of is
How many solid cubes are required to make a solid cube?
Shuxin begins with red
candies, yellow candies, and
blue candies. After eating some
of the candies, there are equal numbers of red, yellow, and blue candies
remaining. What is the smallest possible number of candies that Shuxin
ate?
Square is divided into
smaller congruent squares, as
shown.
What fraction of is
shaded?
How many integers are greater than and less than ?
The line with equation is reflected in the -axis. What is the -intercept of the new line?
If , what is
the value of ?
In the diagram, a semi-circle has centre and diameter . Also, is a point on the circumference such
that .
The measure of is
In a photograph, Aristotle, David, Flora, Munirah, and Pedro are
seated in a random order in a row of chairs. If David is seated in the
middle of the row, what is the probability that Pedro is seated beside
him?
Figure 1 shows an arrangement of lines with intersection point, and Figure 2 shows
an arrangement of lines with
intersection points.
What is the maximum number of intersection points that can appear in
an arrangement of lines?
The numbers , , , , and are rearranged so that the sum of the
first three numbers is equal to the sum of the last three numbers. Which
number is in the middle of this rearrangement?
For some integers and
, the expression is equal to a quadratic
expression in with a constant
term of . Which of the following
cannot be a value of ?
In the diagram, point is
on side of quadrilateral . Also, and are perpendicular to , , , and .
If , what is the
perimeter of quadrilateral ?
Anila’s grandmother wakes up at the same time every day and
follows this same routine:
She gets her coffee hour
after she wakes up. This takes
minutes.
She has a shower hours
after she wakes up. This takes
minutes.
She goes for a walk hours
after she wakes up. This takes
minutes.
She calls her granddaughter hours after she wakes up. This takes
minutes.
She does some yoga hours
after she wakes up. This takes
minutes.
If Anila’s grandmother woke up
minutes ago, what will she be doing in hours?
- Getting her coffee
- Having a shower
- Doing some yoga
- Going for a walk
- Calling her granddaughter
Francesca put one of the integers , , , , , , , ,
in each of the nine squares of the by grid shown. No integer is used twice. She calculated the product of the three
integers in each row and wrote the products to the right of the
corresponding rows. She calculated the product of the integers in each
column and wrote the products below the corresponding columns. Finally,
she erased the integers from the nine squares.
Which integer was in the
square marked ?
Each of the variables ,
, , , and represents a positive integer with the
properties that Which of the variables has the
greatest value?
Suppose that and are real numbers that satisfy the two
equations What is the value of ?
Three standard six-sided dice are rolled. The sum of the three
numbers rolled, , is determined.
The probability that is
closest to
A cylinder contains some water. A solid cone with the same height
and half the radius of the cylinder is submerged into the water until
the circular face of the cone lies flat on the circular base of the
cylinder, as shown.
Once this is done, the depth of the water is half of the height of
the cylinder. If the cone is then removed, the depth of the water will
be what fraction of the height of the cylinder?
(The volume of a cylinder with radius and height is and the volume of a cone with radius and height is .)
A table starts
with every entry equal to and is
modified using the following steps:
(i) adding to all three numbers
in any row;
(ii) adding to all three numbers
in any column.
After step (i) has been used a total of times and step (ii) has been used a
total of times, the table appears
as shown.
What is the value of ?
Four distinct integers ,
, , and are chosen from the set . What is the
greatest possible value of ?
In the diagram, is right-angled at .
Point is on and point is on so that is perpendicular to , , , and . What is the length of ?
The integer is the
smallest positive integer that is a multiple of , has more than positive divisors (including and ), and has fewer than positive divisors (including and ). What is the sum of the digits of
?
A sequence of positive
real numbers, , , , , , satisfies and and for every integer with . For example when , . There are such
sequences. What are the rightmost two digits of ?
Encourage your teacher to register you for Hypatia Contest which will be written in April.