IOQM Previous Papers Sept 28 2025

Let $$𝐴𝐵𝐶𝐷$$ be a quadrilateral in the $$xy-plane$$ with $$𝐴𝐵$$ parallel to $$𝐶𝐷$$ and $$𝐴𝐷 = 𝐵𝐶$$. Suppose $$𝐴 = (0, 0)$$, $$𝐵 = (10, 0)$$, $$𝐶 = (8, 5)$$ and $$𝐷 = (𝑎, 𝑏)$$. Determine the value of $$a^{2}b$$.

A function is defined on the set of positive integers such that if $$𝑛$$ is an odd integer, $$𝑓(𝑛) = 𝑛 − 1$$ and if $$𝑛$$ is an even integer, $$𝑓(𝑛) = n^{2} - 1$$. Determine the sum of all possible values of 𝑛 such that $$𝑓(𝑓(𝑛)) = 99$$.

Find the number of positive integers $$𝑛$$ less than or equal to 100 such that $$𝑛$$ is not divisible by any prime number other than $$2$$ or $$3$$.

The six faces of a cubical die are numbered with $$2^{0},2^{1},2^{2},2^{3},2^{4},2^{5}$$ in such a way that the product of the numbers on any pair of opposite faces is $$2^{5}$$. Two such dice are stacked one on top of another. If $$𝑁$$ is the greatest possible sum of the $$9$$ visible numbers (for all such arrangements of dice), find the sum of the squares of the digits of $$𝑁$$.

Let $$𝑁$$ be the coefficient of $$x^{2025}$$ in the expansion of $$(x+1)(x^{2}+3)(x^{4}+5)(x^{8}+7)....(x^{1024}+21)$$. What is the remainder when $$𝑁$$ is divided by $$100$$?

The sum of four distinct prime numbers is $$240$$. If none of the four primes is greater than $$70$$, what is the smallest of the four numbers?

How many positive integers $$𝑛 ≤ 100$$ are divisible by all positive integers $$𝑖$$ such that $$i^{3} ≤ 𝑛 $$?

Consider a $$2 \times 3$$ rectangle made of $$6$$ unit squares. In how many ways can we fill up the six cells using the numbers $$1, 2, 3, 4, 5, 6,$$ one in each cell, such that any two numbers in adjacent cells (that is, in cells that share a common side) are coprime to each other?

Find the largest integer $$𝑛$$ such that a square of side length $$𝑛$$ is contained in a circular disc of area $$1000$$.

Find the largest positive integer 𝑛 for which the inequality $$\sum_{k=1}^{2n}(-1)^{k}k^{2} < 100$$ holds.

Let $$𝑚$$ be a positive integer satisfying the equation $$5(2m + 1)(2m + 3)(2m + 5) = \overline{ababab}$$ where $$𝑎$$ and $$𝑏$$ represent different digits and $$\overline{ababab}$$ is a six digit number. What is the value of $$𝑚 + 𝑎 + 𝑏$$?

Find the number of ordered pairs $$(𝑚, 𝑛)$$ where $$𝑚$$ and $$𝑛$$ are positive integers less than or equal to $$20000$$ such that $$m^{2} + n^{4}$$ is a power of $$2$$.

In a convex quadrilateral $$𝐴𝐵𝐶𝐷$$, the lengths of the diagonals are $$12$$ and $$16$$ and the line segments joining the midpoints of the opposite sides are of equal length. What is the maximum possible area of the quadrilateral $$𝐴𝐵𝐶𝐷$$?

The side $$𝐴𝐵$$ of a square $$𝐴𝐵𝐶𝐷$$ is $$1$$ and it is also a chord of a circle $$𝑆$$. The side $$𝐶𝐷$$ does not intersect $$𝑆$$. The length of the tangent $$𝐶𝐾$$, drawn from $$𝐶$$ to $$𝑆$$ at the point $$𝐾$$ is $$2$$. If $$𝑑$$ is the diameter of $$𝑆$$, then calculate $$d^{2}$$.

If $$𝑎, 𝑏, 𝑐, 𝑑$$ are positive integers such that $$17(𝑎𝑏𝑐𝑑 + 𝑎𝑏 + 𝑎𝑑 + 𝑐𝑑 + 1) = 20(𝑏𝑐𝑑 + 𝑏 + 𝑑),$$ find $$a^{2}+b^{2}+c^{2}+d^{2}$$.

If $$$1-\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5+\frac{1}{6+\frac{1}{7}}}}}}=\frac{1}{x_1+\frac{1}{x_2+\frac{1}{x_3+\frac{1}{x_4+\frac{1}{x_5+\frac{1}{x_6+\frac{1}{x_7}}}}}}}$$$ where $$x_{1},x_{2},....,x_{7}$$ are positive integers, find $$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}$$.

There are $$100$$ cards in a box which are numbered from $$1$$ to $$100$$. While being blindfolded, Mainak is going to draw one or more cards from the box. After that, he will remove his blindfold and multiply together the numbers on these cards. Mainak wants the product of the numbers on the cards drawn to be a multiple of $$6$$. How many cards does he need to draw to make sure that this will happen?

In the plane let the positive end of the $$𝑥-axis$$ be directed towards East and the positive end of the $$y-axis$$ be directed towards North. Suppose you are at $$(0, 0)$$ and you want to go to $$(7, 12)$$. At every move you are allowed to move unit length towards East or unit length towards North from your current position but you are not allowed to visit any point $$(ℎ, 𝑘)$$ where both $$ℎ, 𝑘$$ are odd. Find the number of such paths $$𝑛$$.

Find the number of ordered pairs $$(𝑚, 𝑛)$$ where $$𝑚$$ and $$𝑛$$ are positive integers such that $$1 ≤ 𝑚 < 𝑛 ≤ 50$$ and the product $$𝑚𝑛$$ is a perfect square.

How many four digit numbers $$\overline{abcd}$$, with non-zero digits $$𝑎, 𝑏, 𝑐, 𝑑$$ in base $$10$$, are there such that $$𝑎 + 𝑐 = 𝑏𝑑$$ and $$𝑏 + 𝑑 = 𝑎𝑐?$$

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function satisfying $$ 4f(3-x)+3f(x)=x^2 $$ for any real $$x$$. Find the value of $$ f(27)-f(25) $$ to the nearest integer.

Three girls $$G_1, G_2, G_3,$$ each read four stories $$ S_1, S_2, S_3, S_4 $$ and discuss which ones they like. No story is liked by all the three. For each of the three pairs of the girls, there is at least one story which is liked by the pair and not liked by the third. Let $$n$$ be the number of ways in which this is possible. Find the sum of the squares of the digits of $$n$$.

Let $$P$$ be a point in the interior of a triangle $$ABC$$ and let $$AP, BP, CP$$ meet the sides $$BC, CA, AB$$ in $$D, E, F$$ respectively. If $$ \frac{BP}{PE}=\frac{5}{2}, \frac{CP}{PF}=\frac{7}{3},$$ and $$ \frac{AP}{PD}=\frac{p}{q}, $$ where $$p$$ and $$q$$ are natural numbers and $$gcd(𝑝, 𝑞) = 1$$, find $$𝑝 + 𝑞$$.

If $$a$$ and $$b$$ are positive integers satisfying $$4^{a}+4a^{2}+4=b^{2}$$, what is the maximum possible value of $$a+b$$?

How many natural numbers $$n\leq 105$$ are there such that $$ 7\mid 2^{n}-n^{2} $$?

Let $$ABC$$ be a triangle, $$D$$ be the midpoint of side $$BC$$, $$O$$ be the circumcenter and $$H$$ be the orthocenter. If the triangle $$ODH$$ is equilateral with side length equal to $$6$$ and the area of the triangle $$ABC$$ can be written as $$a\sqrt{b}$$, where $$a, b$$ are positive integers and $$b$$ is not divisible by the square of any prime, find $$a+ b$$.

Consider the collection $$M$$ of all ordered pairs $$(a,b)$$ of positive integers $$a$$ and $$b$$ which satisfy 

$$ ab=406+11\cdot\operatorname{lcm}(a,b)+7\cdot\gcd(a,b).$$ 

What is the smallest possible value of $$a+b$$?

There are $$10$$ members in a delegation. No two of them have the same height. Let $$N$$ be the number of ways in which they can stand in a line for a photograph such that

1) the leftmost person is the shortest,

2) the rightmost person is the tallest, and

3) in the line between the shortest and tallest person, there is exactly one person who is shorter than both of his immediate neighbours.

If $$N$$ can be written as $$100a+b$$ where $$a$$ and $$b$$ are positive integers less than $$100$$, find $$a+b$$.

Let $$ABC$$ be an isosceles triangle with sides $$13$$, $$13$$ and $$10$$. The tangents to the incircle, drawn parallel to the sides, intersect the sides in points $$D$$, $$E$$, $$F$$, $$G$$, $$H$$, $$K$$ which form a hexagon. If the area of the hexagon $$DEFGHK$$ is $$m+\frac{n}{l}$$, where $$m,n,l$$ are positive integers with $$n<l$$ and $$\gcd(n,l)=1$$, what is $$m+n+l$$?

The vertices of a regular dodecagon (a polygon with $$12$$ sides) are coloured either blue or red. Let $$N$$ be the number of all possible colourings such that no three points of the same colour form the vertices of an equilateral triangle, and no four points of the same colour form the vertices of a square. If $$N$$ can be written as $$N=100p+q$$ where $$p,q$$ are two positive integers less than $$100$$, find $$p+q$$.