IOQM Previous Papers Sept 8 2024-25

The smallest positive integer that does not divide $$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$$ is:

The number of four-digit odd numbers having digits $$1, 2, 3, 4$$, each occuring exactly once, is:

The number obtained by taking the last two digits of $$5^{2024}$$ in the same order is:

Let $$ABCD$$ be a quadrilateral with $$\angle ADC = 70^\circ$$, $$\angle ACD = 70^\circ$$, $$\angle ACB = 10^\circ$$ and $$\angle BAD = 110^\circ$$. The measure of $$\angle CAB$$ (in degrees) is:

Let $$a = \frac{x}{y} + \frac{y}{z} + \frac{z}{x}$$, let $$b = \frac{x}{z} + \frac{y}{x} + \frac{z}{y}$$ and let $$c = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} + \frac{y}{x} + \frac{z}{y} + \frac{x}{z}$$. The value of $$|ab - c|$$ is:

Find the number of triples of real numbers $$(a,b,c)$$ such that $$a^{20}+b^{20}+c^{20}=a^{24}+b^{24}+c^{24}=1$$.

Determine the sum of all possible surface areas of a cube two of whose vertices are $$(1, 2, 0)$$ and $$(3, 3, 2)$$.

Let $$n$$ be the smallest integer such that the sum of digits of $$n$$ is divisible by $$5$$ as well as the sum of digits of $$(n+1)$$ is divisible by $$5$$. What are the first two digits of $$n$$ in the same order?

Consider the grid of points $$X = \{(m, n) : 0 < m, n \le 4\}$$. We say a pair of points $$(a, b), (c, d)$$ in $$X$$ is a knight-move pair if $$(c = a \pm 2 \text{ and } d = b \pm 1)$$ or $$(c = a \pm 1 \text{ and } d = b \pm 2)$$. The number of knight-move pairs in $$X$$ is:

Determine the number of positive integral values of $$p$$ for which there exists a triangle with sides $$a$$, $$b$$, and $$c$$ which satisfy $$a^{2}+(p^{2}+9)b^{2}+9c^{2}-6ab-6pbc=0$$.

The positive real numbers $$a, b, c$$ satisfy: $$\frac{a}{2b+1}+\frac{2b}{3c+1}+\frac{3c}{a+1}=1$$ and $$\frac{1}{a+1}+\frac{1}{2b+1}+\frac{1}{3c+1}=2$$. What is the value of $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$?

Consider a square $$ABCD$$ of side length $$16$$. Let $$E, F$$ be points on $$CD$$ such that $$CE=EF=FD$$. Let the line $$BF$$ and $$AE$$ meet in $$M$$. The area of $$\triangle MAB$$ is:

Three positive integers $$a, b, c$$ with $$a>c$$ satisfy the following equations: $$ac+b+c=bc+a+66$$ and $$a+b+c=32$$. Find the value of $$a$$.

Initially, there are $$3^{80}$$ particles at the origin $$(0, 0)$$. At each step the particles are moved to points above the x-axis as follows: if there are $$n$$ particles at any point $$(x, y)$$, then $$\frac{n}{3}$$ of them are moved to $$(x+1,y+1)$$, $$\frac{n}{3}$$ are moved to $$(x,y+1)$$ and the remaining to $$(x-1,y+1)$$. For example, after the first step, there are $$3^{79}$$ particles each at $$(1, 1)$$, $$(0, 1)$$ and $$(-1, 1)$$. After the second step, there are $$3^{78}$$ particles each at $$(2, 2)$$ and $$(-2, 2)$$, $$2\cdot 3^{78}$$ particles each at $$(1, 2)$$ and $$(-1, 2)$$, and $$3^{79}$$ particles at $$(0, 2)$$. After $$80$$ steps, the number of particles at $$(79, 80)$$ is:

Let $$X$$ be the set consisting of twenty positive integers $$n, n+2,...,n+38$$. The smallest value of $$n$$ for which any three numbers $$a, b, c \in X$$, not necessarily distinct, form the sides of an acute-angled triangle is:

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

Consider an isosceles triangle $$ABC$$ with sides $$BC=30,$$ $$CA=AB=20$$. Let $$D$$ be the foot of the perpendicular from $$A$$ to $$BC$$, and let $$M$$ be the midpoint of $$AD$$. Let $$PQ$$ be a chord of the circumcircle of triangle $$ABC$$, such that $$M$$ lies on $$PQ$$ and $$PQ$$ is parallel to $$BC$$. The length of $$PQ$$ is:

Let $$p, q$$ be two-digit numbers neither of which are divisible by $$10$$. Let $$r$$ be the four-digit number by putting the digits of $$p$$ followed by the digits of $$q$$ (in order). As $$p, q$$ vary, a computer prints $$r$$ on the screen if $$\gcd(p,q)=1$$ and $$p+q$$ divides $$r$$. Suppose that the largest number that is printed by the computer is $$N$$. Determine the number formed by the last two digits of $$N$$ (in the same order).

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, so that no three of the points form a triangle with lines of the same color.

On a natural number $$n$$ you are allowed two operations: (1) multiply $$n$$ by $$2$$ or (2) subtract $$3$$ from $$n$$. For example starting with $$8$$ you can reach $$13$$ as follows: $$8\rightarrow 16\rightarrow 13$$. You need two steps and you cannot do in less than two steps. Starting from $$11$$, what is the least number of steps required to reach $$121$$?

An integer $$n$$ is such that $$\frac{n-9}{4}$$ is a three digit number with equal digits, and $$\frac{n-172}{9}$$ is a $$4$$ digit number with the digits $$2, 0, 2, 4$$ in some order. What is the remainder when $$n$$ is divided by $$100$$?

In a triangle $$ABC, \angle BAC = 90^\circ$$. Let $$D$$ be the point on $$BC$$ such that $$AB + BD = AC + CD$$. Suppose $$BD : DC = 2 : 1$$. If $$\frac{AC}{AB} = \frac{m+\sqrt{p}}{n}$$, where $$m, n$$ are relatively prime positive integers and $$p$$ is a prime number, determine the value of $$m + n + p$$.

Consider the fourteen numbers, $$1^4, 2^4, \dots, 14^4$$. The smallest natural number $$n$$ such that they leave distinct remainders when divided by $$n$$ is:

Consider the set $$F$$ of all polynomials whose coefficients are in the set of $$\{0, 1\}$$. Let $$q(x) = x^3 + x + 1$$. The number of polynomials $$p(x)$$ in $$F$$ of degree $$14$$ such that the product $$p(x)q(x)$$ is also in $$F$$ is:

A finite set $$M$$ of positive integers consists of distinct perfect squares and the number $$92$$. The average of the numbers in $$M$$ is $$85$$. If we remove $$92$$ from $$M$$, the average drops to $$84$$. If $$N^{2}$$ is the largest possible square in $$M$$, what is the value of $$N$$?

The sum of $$[x]$$ for all real numbers $$x$$ satisfying the equation $$16+15x+15x^{2}=[x]^{3}$$ is:

In a triangle $$ABC$$, a point $$P$$ in the interior of $$ABC$$ is such that $$\angle BPC-\angle BAC=\angle CPA-\angle CBA=\angle APB-\angle ACB$$. Suppose $$\angle BAC=30^{\circ}$$ and $$AP=\sqrt{m}$$. Let $$D, E, F$$ be the feet of perpendiculars from $$P$$ to $$BC, CA, AB$$, respectively. If $$\frac{m}{n}$$ is the area of the triangle $$DEF$$ where $$m, n$$ are integers with $$n$$ prime, then what is the value of the product $$mn$$?

Find the largest positive integer $$n<30$$ such that $$\frac{1}{2}(n^{8}+3n^{4}+4)$$ is not divisible by the square of any prime number.

Let $$n=2^{19}3^{12}$$. Let $$M$$ denote the number of positive divisors of $$n^{2}$$ which are less than $$n$$ but would not divide $$n$$. What is the number formed by taking the last two digits of $$M$$ (in the same order)?

Let $$ABC$$ be a right-angled triangle with $$\angle B=90^{\circ}$$. Let the length of the altitude $$BD$$ be equal to $$12$$. What is the minimum possible length of $$AC$$, given that $$AC$$ and the perimeter of triangle $$ABC$$ are integers?