2021 HMC 10
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
Let ,
and
. The value of
can be expressed as
in the simplest form. Find
.
Problem 2
James is standing in the origin of a plane. He walks units to the south, and turns
to the left, walked
units. Find the distance between his position now and the origin.
Problem 3
In Martian civilization, when they are doing addition, in their perspective, . Which of the following ordered pair of
would result in this rule to be true?
Problem 4
There exists a rectangular prism such that the sum of all the side lengths is equal to units and the total surface area is
units. Find the length of a space diagonal of this rectangular prism.
Problem 5
Let be a triangle with side length
. The angle bisector of
intersects
at
. Find
.
Problem 6
Let and let
. Find the value of
.
Problem 7
Dr.Jeck is arranging books on a bookshelf. How many ways are there to arrange two identical math books, three identical science books and four identical English books such that the first and last books must be English books?
Problem 8
Let and let
be the
th triangular number. Find
Problem 9
Kashuv won a game, and he wanted to inform his friends about it. On the first day, he told his three best friends about the game. For each consecutive days, those who were informed about the victory of the game would introduce to four more best friends. Find the total number of people informed about the game on the th day assume that no people can be informed twice.
Problem 10
Let and
. If
and
are both nonzero integers, how many possible ordered pairs
would result in
and
has exactly one common root?
Problem 11
Let be a sequence of real numbers such that
and
. How many terms from
to
, inclusive, are positive integers?
Problem 12
Joana is walking on a straight road that goes from south to north. Assume that all of his step is straight in the direction of north. Joana started from a point, walked to north, and then he decided to walk backward
. After, he decided to walk
, and walks backward
. Then, he walked forward
and walked backward
. Continuing on this sequence that he walks
meter forward
th time and then walks half the distance he walked forward immediately after that. Find the distance between his position after the
nd time he walked backward and
time he walked forward.
Problem 13
can be expressed as
. Find
Problem 14
Let be a polynomial with degree of
such that
and the coefficient of the
term is equal to the reciprocal of the constant term. What is the smallest possible value of
?
Problem 15
Let . Find the number of positive integer factors of
.
Problem 16
In the large square with side length , a quarter circle arc of radius
intersects two sides that divided the side length into ratio
, as shown in the diagram. Find the area of the shaded region.
Problem 17
The largest root of the polynomial can be expressed as
. Find
.
Problem 18
Jonas has a collection of toys, and he wanted to organize them into boxes. The collection contains toys, and he wants to put them into
different boxes, each numbered from
to
, respectively. Also, each boxes can only contain one toy. However, due to his believe of equity, he decided to do a paper lottery by putting twenty one sheets of paper that represents each positive integer from
to
, inclusive. After he randomly choose any sheet, he would take out and not replace back into the cup. The number on the sheet he chose for a specific toy would determine the box that the toy would stay in. Jonas first rearranged all the toys and started this procedure. What is the probability that toy
would be in box
?
Problem 19
Find the largest positive integer less than or equal to that divides
.
Problem 20
Let be the area enclosed by the graph of equation
. Find
Problem 21
Joshua wrote all the divisors of , all the divisors of
, and all the divisors of
on different sheets of paper, such that only
number is on each sheets. Then, he putted all the sheets into a jar and randomly choose
sheets from the jar. Find the probability that the two sheets contains the same number.
Problem 22
Let . Let
be the roots of
. Furthermore, let
. Find the value of
Problem 23
Let be a monic polynomial with degree
such that
is a root of
and all the coefficients of
are rational numbers. Find
Problem 24
At Janus high school, in a math class, students are playing an icebreaker game. They each wrote a name tag that has their name, and each names are different. Then, they gave their name tag to the teacher, and the teacher rearranged them and randomly give each student a name tag after rearrangement. Given that there are students in that class, find the expected value of the number of students who get their own name tag back.
Problem 25
In the following diagram, the three concentric squares have areas ,
,
and
units, respectively. The smallest square was formed by rotating one of the larger squares by
degrees. John has a uniform circular coin with radius
, and he wants to place it in the sheet of three concentric squares such that a part of the coin lands in the sheet(Not necessarily the entire coin). To do that, he would first randomly choose and label a point for the coin's center(Possibly outside this diagram), and then he would place the coin. The probability that the coin would intersects this diagram at two points can be expressed as
Find
.