1. Let n be a two-digit integer, and let m be the result when we reverse the digits of n.

If n − m
and n + m are both perfect squares, find n.

2. Compute the number of ordered triples (p, q, r) of primes, each at most 30, such that

p + q + r = p2 + 4

3. Let 0◦ < θ < 90◦ be an angle. If

logsin θ cos θ, logcos θ tan θ, logtan θ sin θ

√
a−b
form a geometric progression in that order, let sin θ = c where a, b, c are relatively prime positive
integers. Find a + b + c.

4. Inside a bag containing geometric shapes, there are twice as many equilateral triangles
with side l as squares with side 3. If the sum of the areas of all the squares is equal to the sum
of the areas of all the equilateral triangles, find the integer nearest to l.

5. Let △ABC be an equilateral triangle, and let D be a point √

such that B is the midpoint of
AD. If CD = 12, the area of △ABC can be expressed in the form a b for some integers a, b, where
b is not divisible by the square of any prime. Find a + b.

6. There exists unique digits a ̸= 0 and b ̸= a such that the fourdigit number aabb is a perfect
square. Compute a + b.

7. How many ordered pairs (x, y) of integers satisfy

x4 + y 4 + 2x2 y 2 + 1023 = 1024x2 + 1024y 2 ?

8. Let p, q, and r be distinct prime numbers such that p2 + 1, q 2 + 2, and r2 + 3, form an arith-
metic sequence in that order. What is the minimum possible value of p + q + r ?

9. Arrange the numbers 0, 1, 2, . . . , 23 in a circle. Let the expected number of (unordered) pairs
of adjacent numbers that sum to 23 be m n where m, n are relatively prime positive integers. Find
m + n.

10. Three triplets, Tam, Tom, and Tim, work together to build a treehouse. Tam works twice
as fast as Tom, but three times slower than Tim. After working together for 10 days, Tim decides
to quit since he is tired. After another 10 days, Tom also decides to quit. It takes Tam another
6 days to complete the treehouse. The number of days it would had taken them to complete the
treehouse if none of them had quit can be written as m n for relatively prime positive integers m
and n. What is m + n ?

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 11. Let ABCD be a parallelogram with area 2020 such that AB/BC = 66/65. The bisectors of
∠DAB, ∠ABC, ∠BCD, ∠CDA form a quadrilateral. Let the area of this quadrilateral be mn where
m, n are relatively prime positive integers. Find the integer nearest to (m + n)/10

12. Let ABC be a triangle with incircle ω. Points E, F lie on AB, AC such that EF ∥BC and EF
is tangent to ω. If EF = 6 and BC = 7, find AB + AC.

13. Define f (n) to be the sum of the positive integer factors of n. If f (n) = 360 and f (3n) = 1170,
what is the sum of square of digits of the smallest possible value of f (2n) ?

14. Call a positive integer super-even if the last two digits of all its multiples are always even.
If S is the sum of all super-even integers 1 ≤ n ≤ 2018, what is the number of factors of S?

15. How many integers 1 ≤ m ≤ 100 are there such that there exists an integer n which has
the property that exactly m% of the positive integer divisors of n are perfect squares?

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 16. Let ABCD be a quadrilateral with side lengths AB = 2, BC = 5, CD = 3, and suppose
∠B = ∠C = 90◦ . Let M be the midpoint of AD and
√
let P be a point on BC so that quadrilaterals
ABP M and DCP M have equal areas. Let P M be ba where a, b are positive integers such that a is
square-free. Find a + b.

17. There is a unique nondecreasing sequence of positive integers a1 , a2 , . . . , an such that

    
1 1 1
a1 + a2 + · · · an + = 2020.
a1 a2 an

Compute a1 + a2 + · · · + an .

18. In △ABC, AC = 3, AB = 4, BC = 5, and I is the incenter of the triangle. Extend BI and CI

to points D and E, respectively, such that BE ⊥ BI and CD ⊥ CI. What is the integer nearest to
the ratio between the areas of △BEI and △CDI ?

19. A positive integer is called triangular number if it can be expressed in the form n(n+1)
2 for
a positive integer n. How many ordered pairs (x, y) of triangular numbers satisfy x − y = 31, 500 ?

20. Consider quadrilateral ABCD with AB = 3, BC = 4, CD = 5, and DA = 6. Circle Γ

passes through A, B, C and intersects DA again at X. Given that CX = 4, compute the area of
quadrilateral ABCD

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 21. In a plane lies 14 points and all possible lines are drawn among themselves, forming 287
triangles. Suppose N quadrilaterals are formed. Find the sum of square of digits of N .

22. Each of the six boxes shown in the equation below is replaced with a distinct number
chosen from {1, 2, 3, . . . , 15}.
□ □ □
S= + + .
□ □ □
Suppose that the order of the fractions doesn’t matter. Then there is exactly one way to arrange
six numbers into the boxes such that S < 1 and S is as large as possible. Compute the sum of
the 6 numbers.

23. Compute the maximum value of n for which n cards, numbered 1 through n, can be
arranged and lined up in a row such that
• it is possible to remove 20 cards from the original arrangement leaving the remaining cards
in ascending order, and
• it is possible to remove 20 cards from the original arrangement leaving the remaining cards
in descending order.

24. Let ABC be a triangle and let M be the midpoint of BC. The lengths AB, AM, AC form a
geometric sequence in that order. The side lengths of △ABC are 2020, 2021, x in some order. Let
the sum of all possible values of x be S. Find the last two digits of the integer nearest to S.

25. Let ⌈x⌉ denote the smallest integer greater than or equal to x. The sequence (ai ) is defined
as follows: a1 = 1, and for all i ≥ 1,
    
ai + 1 ai + 1
ai+1 = min 7 , 19 .
7 19

Find the last two digits of a100 .

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 26. Miriam is trying to escape from a haunted house placed in the middle of a circular field
with area 100 square feet. Each minute, she carefully moves 1 foot in any direction she wants;
however, an evil ghost may decide to scare her and make her run 1 foot in the direction opposite
to the one she wanted. What are the last two digits of the least integer number of feet Miriam
must walk in order to escape from the field, independently from the ghost’s tricks?

27. Eddie has a birthday party, in which he invited his 8, very generous, best friends. Each
friend brought a certain number of gifts for Eddie. It is observed that for every group of 4 friends,
at least 2 of them brought the same number of gifts. Moreover, the person with the most gifts
brought 4, while the least brought 1. What is the total number of possible values of the amount
of gifts Eddie received in total?

28. In △ABC, AB = 4, BC = 5, AC = 6, and I is the incenter of the triangle. Let S be the locus
of all points D such that the circumcenter O of △ABD is collinear with AI. Determine the largest
integer less than or equal to the perimeter of S.

29. The AMC 12 consists of 25 problems, where for each problem, a correct answer is worth 6
points, leaving the problem blank is worth 1.5 points, and an incorrect answer is worth 0 points.
The AIME consists of 15 problems, where each problem is worth 10 points, and no partial credit
is given.
Any contestant who scores at least 84 on the AMC 12 is eligible for the AIME, and the USAMO
index of such a student is the sum of his AMC 12 and AIME scores. Determine the last two digits
of the smallest integer N > 300 such that no contestant can possibly obtain a USAMO index of 12 N .

30. Let P0 , P1 , . . . be points in the complex plane such that P0 = −1, P1 = i and Pn = i·Pn−1 +Pn−2
for every n ≥ 2. Let P be the polygon P0 P1 . . . P2017 and let √
P√and A be its perimeter and area,
P a b+c d
respectively. If the value of A can be written in the form e for positive integers a, b, c, d, and
e such that b and d are not divisible by the square of any prime and gcd(a, c, e) = 1, compute the
remainder when a + b + c + d + e is divided by 100.

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