Algebra 10
Geometry 10
Algebra 12
Geometry 12
Counting and Probability
100
What is the value of 
The likely fastest method will be direct computation. 
evaluates to 
and 
evaluates to 
. The difference is 
100
A rectangle has integer side lengths and an area of 
. What is the least possible perimeter of the rectangle?
We can start by assigning the values x and y for both sides. Here is the equation representing the area:
Let's write out 2024 fully factorized.
Since we know that 
, we want the two closest numbers possible. After some quick analysis, those two numbers are 
and 
. 
Now we multiply by 
and get 
100
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
If the person is the 1015th from the left, that means there is 1014 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are 
people in line.
100
A square and an isosceles triangle are joined along an edge to form a pentagon 
inches tall and 
inches wide, as shown below. What is the perimeter of the pentagon, in inches?
Drop an altitude from the vertex of the isosceles triangle to the midpoint of the base, thereby creating two right triangles whose legs are 
and 
. It follows that the two congruent sides have length 
, hence, the perimeter of the pentagon is 
.
100
Janet rolls a standard 
-sided die 
times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 
?
There are cases where the running total will equal : one roll; two rolls; or three rolls:
Case 1: The chance of rolling a running total of , namely 
in exactly one roll is 
.
Case 2: The chance of rolling a running total of in exactly two rolls, namely 
and 
is 
.
Case 3: The chance of rolling a running total of 3 in exactly three rolls, namely 
is 
.
Using the rule of sum we have 
200
Integers 
, 
, and 
satisfy 
, 
, and 
. What is 
Subtracting the first two equations yields 
. Notice that both factors are integers, so 
could equal one of 
and 
. We consider each case separately: 3 For 
, from the second equation, we see that 
. Then 
, which is not possible as is an integer, so this case is invalid.
For 
, we have 
and 
, which by experimentation on the factors of 
has no solution, so this is also invalid.
For 
, we have 
and 
, which by experimentation on the factors of has no solution, so this is also invalid.
Thus, we must have 
, so 
and 
. Thus 
, so 
. We can simply trial and error this to find that 
so then 
. The answer is then 
.
200
AMC 2024 10B –11
In the figure below 
is a rectangle with 
and 
. Point 
lies on 
, point 
lies on 
, and 
is a right angle. The areas of 
and 
are equal. What is the area of 
?
We know that 
, 
, so 
and 
. Since 
, triangles 
and 
are similar. Therefore, 
, which gives 
. We also know that the areas of triangles and 
are equal, so 
, which implies 
. Substituting this into the previous equation, we get 
, yielding 
and 
. Thus,
200
In the following expression, Melanie changed some of the plus signs to minus signs:
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
Recall that the sum of the first 
odd numbers is 
. Thus
If we want to minimize the number of sign flips to make the number negative, we must flip the signs corresponding to the values with largest absolute value. This will result in the inequality
The positive section of the sum will contribute , and the negative section will contribute 
. The inequality simplifies to
The greatest positive value of satisfying the inequality is 
, corresponding to 
positive numbers, and 
negatives.
200
Cyclic quadrilateral 
has lengths 
and 
with 
. What is the length of the shorter diagonal of ?
~diagram by erics118
First, 
by properties of cyclic quadrilaterals.
Let 
. Apply the Law of Cosines on 
:
Let 
. Apply the Law of Cosines on 
:
By Ptolemy’s Theorem,
Since 
, The answer is 
.
200
A group of 
students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students and 
, student speaks some language that student does not speak, and student speaks some language that student does not speak. What is the least possible total number of languages spoken by all the students?
Let's say we have some number of languages. Then each student will speak some amount of those languages, and no two people can have the same combination of languages or else the conditions will no longer be satisfied. Notice that 
. So each of the students can speak some of the 
languages. Thus, 
is our answer.
300
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only 
full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
We let 
denote how much juice we take from each of the first children and give to the th child.
We can write the following equation: 
, since each value represents how much juice each child (equally) has in the end. (Each of the first three children now have 
juice, and the fourth child has 
more juice on top of their initial 
.)
Solving, we see that 
300
All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length 
? 
Using the rectangle with area 
, let its short side be and the long side be 
. Observe that for every rectangle, since ratios of the side length of the rectangles are directly proportional to the ratios of the square roots of the areas (For example, each side of the rectangle with area is 
times that of the rectangle with area ), as they are all similar to each other.
The side opposite on the large rectangle is hence written as 
. However, can be written as 
. Since the two lengths are equal, we can write 
, or 
. Therefore, we can write 
.
Since 
, we have 
, which we can evaluate as 
. From this, we can plug back in to to find 
. Substituting into , we have 
which can be evaluated to 
.
300
How many positive perfect squares less than 
are divisible by 
?
Since is square-free, each solution must be divisible by 
. We take 
and see that there are 
positive perfect squares no greater than 
.
300
In the figure below is a rectangle with and . Point lies , point lies on , and is a right angle. The areas of and 
are equal. What is the area of ?
Note: On certain tests that took place in China, the problem asked for the area of 
.
We know that , , so and . Since , triangles and are similar. Therefore, , which gives . We also know that the areas of triangles and are equal, so , which implies . Substituting this into the previous equation, we get , yielding and . Thus,
300
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
Let's say we're dealing with the following snails:A,B,C,D,E.
5 snails tied: All 5 snails tied for 1st place, so only 1way.
snails tied: A,B,C,D all tied, and E either got 1st or last. 
ways to choose who isn't involved in the tie and ways to choose if that snail gets first or last, so ways.
3 snails tied: We have ABC,D,E. There are 
ways to determine the ranking of the groups. There are 
ways to determine the two snails not involved in the tie. So 
ways.
2 snails tied: We have AB,C,D,E. There are 
ways to determine the ranking of the groups. There are 
ways to determine the three snails not involved in the tie. So 
ways.
1 snail tied: This is basically just every snail for a place, so 
ways.
The answer is 
.
400
What is the remainder when 
is divided by 
?
Completing the square, then difference of squares:
Thus, we see that the remainder is 
400
Let 
be the kite formed by joining two right triangles with legs and 
along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle 
?
Let be quadrilateral 
. Drawing line 
splits the triangle into 
. Drawing the altitude from 
to point 
on line , we know 
is 
, 
is 
, and 
is 
.
Due to the many similarities present, we can find that is 
, and the height of is 
is 
and the height of is 
.
Solving for the area of gives 
which is 
400
What is the value of
To solve this problem, we will be using difference of cube, sum of squares and sum of arithmetic sequence formulas.
we could rewrite the second part as 
Hence,
Adding everything up:
400
Equilateral with side length 
is rotated about its center by angle 
, where 
, to form 
. See the figure. The area of hexagon 
is 
. What is 
?
Let O be circumcenter of the equilateral triangle
Easily get 
is invalid given 
, 
400
Suppose that cards numbered 
are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 
are picked up on the first pass, and on the second pass, on the third pass, 
on the fourth pass, and 
on the fifth pass. For how many of the 
possible orderings of the cards will the cards be picked up in exactly two passes?
For 
suppose that cards 
are picked up on the first pass. It follows that cards 
are picked up on the second pass.
Once we pick the spots for the cards on the first pass, there is only one way to arrange all 
cards.
For each value of 
there are 
ways to pick the 
spots for the cards on the first pass: We exclude the arrangement
in which the cards are arranged such that the first pass consists of all cards.
Therefore, the answer is
500
Hiram's algebra notes are 
pages long and are printed on 
sheets of paper; the first sheet contains pages and , the second sheet contains pages and , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 
. How many sheets were borrowed?
Suppose the roommate took sheets through , or equivalently, page numbers 
through 
. Because there are 
numbers taken,
The first possible solution that comes to mind is if 
, which indeed works, giving 
and 
. The answer is 
500
Each of 
bricks (right rectangular prisms) has dimensions 
, where , , and are pairwise relatively prime positive integers. These bricks are arranged to form a 
block, as shown on the left below. A 
th brick with the same dimensions is introduced, and these bricks are reconfigured into a 
block, shown on the right. The new block is unit taller, unit wider, and unit deeper than the old one. What is 
?
The xx block has side lengths of 
. The xx
block has side lengths of 
.
We can create the following system of equations, knowing that the new block has unit taller, deeper, and wider than the original:
Adding all the equations together, we get 
. Adding 
to both sides, we get 
. The question states that 
are all relatively prime positive integers. Therefore, our answer must be congruent to 
. The only answer choice satisfying this is 
.
500
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers in the 2023rd row?
First, let 
be the sum of the th row. Now, with some observation and math instinct, we can guess that 
.
Now we try to prove it by induction,
(works for base case)
By definition from the question, the next row is always
Double the sum of last row (Imagine each number from last row branches off toward left and right to the next row), plus # of new row, minus 2 (minus leftmost and rightmost's 1)
Which gives us
Hence, proven
Last, simply substitute 
, we get 
Last digit of 
is 
, 
500
Suppose , , and 
are points in the plane with 
and 
, and let be the length of the line segment from to the midpoint of 
. Define a function 
by letting 
be the area of . Then the domain of is an open interval 
, and the maximum value 
of occurs at 
. What is 
?
Let midpoint of 
as , extends 
to 
and 
,
triangle 
has sides 
, based on triangle inequality,
so
so 
which is achieved when 
, then 
500
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin 
is 
for 
More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is 
where 
and 
are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 
and 
) What is 
"Evenly spaced" just means the bins form an arithmetic sequence.
Suppose the middle bin in the sequence is . There are 
different possibilities for the first bin, and these two bins uniquely determine the final bin. Now, the probability that these bins are chosen is 
, so the probability is the middle bin is 
. Then, we want the sum
The answer is 