IOQM Mock Test Series 2025-26

for Senior Test Solutions 5

July 9, 2025

Instructions
1. Questions 1 to 10 carry 2 marks each:
Questions 11 to 20 carry 3 marks each:
Questions 20 to 30 carry 5 marks each.
2. All questions are compulsory.
3. There are no negative marks.

4. All answers will be from the set {00, 01, 02, 03, · · ·, 99}
5. Total Marks 100 time 3hrs

1 2M Questions

1. Positive integers a, b, c ≥ 2 satisfy the equation abc + ab + a = 64. Find

the number of different values of a + b + c.

1
2. Suppose for real numbers x1 , x2 , . . . x2018 the following equation holds:
√ √ √ 1
x1 − 1 + x2 − 1 + · · · + x2018 − 1 = (x1 + x2 + · · · + x2018 )
2
Find the sum of the digits of the value of x1 + x2 + · · · + x2018 .
3. The figure below contains two 3−4−5 right triangles attached to a square
of side length 5 . The value of the greatest integer less than or equal to
tan ∠ABC .

4. If x11 + x7 + x3 = 1. and xα = x4 + x3 − 1. Find the minimum value

of α.
5. A large number of children sit in a circle. They count the numbers 1, 2,
3, . . . in clockwise order, starting with the oldest child who counts the
number “1.” Once any child counts a number which is not divisible by 2, 3,
or 5, that child leaves the circle and the next child continues with the next
number. In particular, the oldest child leaves the circle after counting the
number “1.” They count until only one child remains, at which point they
stop counting. Given that the number “121” was the last number counted,
how many children were originally in the circle?
6. Let n be the smallest positive integer with the property that lcm(n, 2020!) =
2021!, where lcm(a, b) denotes the least common multiple of a and b. Let
τ (n) denotes the number of positive factors of n . Find τ (τ (n)).

7. Let A and B be two non-empty subsets of X = {1, 2, ..., 8} with A∪B = X

and A ∩ B = ∅. Let PA be the product of all elements of A and let PB be
the product of all elements of B. Find sum of the digits of the minimum
possible value of sum PA + PB .
8. 62. [Ray Li] In triangle ABC, AB = 36, BC = 40, CA = 44. The bisector
of angle A meet BC at D and the circumcircle at E different from A. Cal-
culate the value of [DE], where [x] is greatest integer less than or equal
to x.

2
 9. Given a rectangle ABCD. Points E and F lie on sides BC and CD re-
spectively so that the area of triangles ABE, ECF , F DA is equal to 1.
Determine the square of the area of triangle AEF .

10. On a given circle, six points A, B, C, D, E, and F are chosen at random,

independently and uniformly with respect to arc length. Let T be the
number of arrangements such that the two triangles ABC and DEF are
disjoint. Find the product of the nonzero digits of T.

2 3M Questions

1. A computer screen shows a 98 × 98 chessboard, colored in the usual way.

One can select with a mouse any rectangle with sides on the lines of the
chessboard and click the mouse button: as a result, the colors in the
selected rectangle switch (black becomes white, white becomes black).
Find, with proof, the minimum number of mouse clicks needed to make
the chessboard all one color.


2. Let f 1 (x) = x2 − 20 for all real numbers x, and let f k (x) = f 1 f k−1 (x)
for all integers k ≥ 2. Let x0 and x1 be the smallest and largest real
solutions to the equation f 2020 (x) = 0, respectively. What is the largest
integer less than or equal to x20 + x21 ?
3. In the following diagram, a semicircle is folded along a chord AN and
intersects its diameter M N at B. Suppose M B : BN = 2 : 3 and M N =
10. If AN = x, find x2 .

3
 4. Find the number of triples of positive integers (x, y, z) that satisfy the
equation

2(x + y + z + 2xyz)2 = (2xy + 2yz + 2zx + 1)2 + 2023.

5. How many ways can six people of different heights stand in line such that
for all 1 ≤ k ≤ 6, the k th tallest person must stand next to either the
(k + 1)th or (k − 1)th tallest person (or both)? In particular, the tallest
person must stand next to the second tallest person, and the shortest
person must stand next to the second shortest person.
6. For any natural number, let’s call the numbers formed from its digits and
have the same "digit" arrangement with the initial number as the "partial
numbers". For example, the partial numbers of 149 are 1, 4, 9, 14, 19, 49, 149,
and the partial numbers of 313 are 3, 1, 31, 33, 13, 313. Find the number of
all natural numbers whose partial numbers are all prime.
7. Find the number of tuples 4 consecutive even integers, such that the sum
of their squares divides the square of their product.
8. Determine number of pairs (k, n) of positive integers that satisfy

1! + 2! + ... + k! = 1 + 2 + ... + n.

9. Find the number of triples of

y 2 − x2 z2 − y2 x2 − z 2
+ + <0
2x2 + 1 2y 2 + 1 2z 2 + 1
where x, y and z are all prime numbers

10. The numbers 1, 2, ..., 2023 are written on the board in this order. We
can repeatedly perform the following operation with them: We select any
odd number of consecutively written numbers and write these numbers in
reverse order. Let P be the number of different possible orders of these
2023 numbers . Given that P = m(n!)2 where m is as small as possible
then find S(m) + S(n) where S(n) denotes the sum of the digits of n.

3 5M Questions

1. Let S(n) denote the sum of all digits of natural number n. Determine
the number of natural numbers n for which both numbers n + S(n) and
n − S(n) are square powers of non-zero integers.
2. Let ABCD be an isosceles trapezoid such that AB = 10, BC = 15, CD =
28, and DA = 15. There is a point E such that △AED and △AEB have
the same area and such that EC is minimal. Find the greatest integer less

4
 than or equal to EC.

3. Reshma makes a large necklace from beads labeled 290, 291, . . . , 2023. She
uses each bead exactly once, arranging the beads in the necklace any order
she likes. Find the number of permutation in which no three beads in a
row have labels are the side lengths of a triangle.
4. [AIME 1986] In △ABC, AB = 425, BC = 450, and AC = 510. An interior
point P is then drawn, and segments are drawn through P parallel to the
sides of the triangle. If these three segments are of an equal length d, find
[ d6 ]where [x] =greatest integer less than or equal to x.

5. Let Y be the smallest positive integer n such that if n squares of a 1000 ×

1000 chessboard are colored, then there will exist three colored squares
whose centers form a right triangle with sides parallel to the edges of the
board. Find the square root of product of nonzero digits of Y.

6. In triangle ABC the medians AD and CE have lengths 18 and 27 , respec-

tively, and AB = 24. Extend CE √ to intersect the circumcircle of ABC at
F . The area of triangle AF B is m n, where m and n are positive integers
and n is not divisible by the square of any prime. Find m + n.

5
 7. Let M be a set composed of n elements and let P(M ) be its power set.
Find the number of functions f : P(M ) → {0, 1, 2, . . . , n} that have the
properties
(a) f (A) ̸= 0, for A ̸= ϕ;
(b) f (A ∪ B) = f (A ∩ B)+f (A∆B), for all A, B ∈ P(M ), where A∆B =
(A ∪ B) \ (A ∩ B).
8. Let 1 ≤ r ≤ n and consider all subsets of r elements of the set {1, 2, . . . , n}.
Each of these subsets has a smallest member. Let F (n, r) denote the
arithmetic mean of these smallest numbers; Find the number of r such
that F (10, r) ∈ N
9. Let A0 = (0, 0). Distinct points A1 , A2 , . . √
. lie on the x-axis, and distinct
points B1 , B2 , . . . lie on the graph of y = x. For every positive integer
n, An−1 Bn An is an equilateral triangle. What is the least n for which the
length A0 An ≥ 100?

10. Let m, n ⊂ R and

f (m, n) = m4 8 − m4 + 2m2 n2 12 − m2 n2 + n4 18 − n4 − 100

Find the smallest possible value for a in which f (m, n) ≤ a, regardless of

the input of f .