IOQM Previous Papers sept 4 2022-23

A triangle $$ABC$$ with $$AC=20$$ is inscribed in a circle $$\omega$$. A tangent $$t$$ to $$\omega$$ is drawn through $$B$$. The distance of $$t$$ from $$A$$ is $$25$$ and that from $$C$$ is $$16$$. If $$S$$ denotes the area of the triangle $$ABC$$, find the largest integer not exceeding $$S/20$$.

In a parallelogram $$ABCD$$, a point $$P$$ on the segment $$AB$$ is taken such that $$\frac{AP}{AB}=\frac{61}{2022}$$ and a point $$Q$$ on the segment $$AD$$ is taken such that $$\frac{AQ}{AD}=\frac{61}{2065}$$. If $$PQ$$ intersects $$AC$$ at $$T$$, find $$\frac{AC}{AT}$$ to the nearest integer.

In a trapezoid $$ABCD$$, the internal bisector of angle $$A$$ intersects the base $$BC$$ (or its extension) at the point $$E$$. Inscribed in the triangle $$ABE$$ is a circle touching the side $$AB$$ at $$M$$ and side $$BE$$ at the point $$P$$. Find the angle $$DAE$$ in degrees, if $$AB:MP=2$$.

Starting with a positive integer $$M$$ written on the board, Alice plays the following game: in each move, if $$x$$ is the number on the board, she replaces it with $$3x+2$$. Similarly, starting with a positive integer $$N$$ written on the board, Bob plays the following game: in each move, if $$x$$ is the number on the board, he replaces it with $$2x+27$$. Given that Alice and Bob reach the same number after playing $$4$$ moves each, find the smallest value of $$M+N$$.

Let $$m$$ be the smallest positive integer such that $$m^{2}+(m+1)^{2}+\cdot\cdot\cdot+(m+10)^{2}$$ is the square of a positive integer $$n$$. Find $$m+n$$.

Let $$a, b$$ be positive integers satisfying $$a^{3}-b^{3}-ab=25$$. Find the largest possible value of $$a^{2}+b^{3}$$.

Find the number of ordered pairs $$(a,b)$$ such that $$a, b\in\{10,11,\cdot\cdot\cdot,29,30\}$$ and $$\gcd(a,b)+\operatorname{lcm}(a,b)=a+b$$.

Suppose the prime numbers $$p$$ and $$q$$ satisfy $$q^{2}+3p=197p^{2}+q$$. Write $$\frac{q}{p}$$ as $$l+\frac{m}{n}$$ where $$l, m, n$$ are positive integers, $$m<n$$ and $$\gcd(m,n)=1$$. Find the maximum value of $$l+m+n$$.

Two sides of an integer sided triangle have lengths $$18$$ and $$x$$ where $$x<100$$. If there are exactly $$35$$ possible integer values $$y$$ such that $$18, x, y$$ are the sides of a non-degenerate triangle, find the number of possible integer values $$x$$ can have.

Consider the $$10$$-digit number $$M=9876543210$$. We obtain a new $$10$$-digit number from $$M$$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $$M$$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $$M=9876543210$$, by interchanging the $$2$$ underlined pairs, and keeping the others in their places, we get $$M_{1}=9786453210$$. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $$M$$.

Let $$AB$$ be a diameter of a circle $$\omega$$ and let $$C$$ be a point on $$\omega$$, different from $$A$$ and $$B$$. The perpendicular from $$C$$ intersects $$AB$$ at $$D$$ and $$\omega$$ at $$E(\neq C)$$. The circle with centre at $$C$$ and radius $$CD$$ intersects $$\omega$$ at $$P$$ and $$Q$$. If the perimeter of the triangle $$PEQ$$ is $$24$$, find the length of the side $$PQ$$.

Given $$\triangle ABC$$ with $$\angle B=60^{\circ}$$ and $$\angle C=30^{\circ}$$, let $$P, Q, R$$ be points on sides $$BA, AC, CB$$ respectively such that $$BPQR$$ is an isosceles trapezium with $$PQ||BR$$ and $$BP=QR$$. Find the minimum possible value of $$\frac{2[ABC]}{[BPQR]}$$ where $$[S]$$ denotes the area of any polygon $$S$$.

Let $$ABC$$ be a triangle and let $$D$$ be a point on the segment $$BC$$ such that $$AD=BC$$. Suppose $$\angle CAD=x^{\circ}$$, $$\angle ABC=y^{\circ}$$ and $$\angle ACB=z^{\circ}$$ and $$x, y, z$$ are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of $$\angle ABC$$ in degrees.

Let $$x, y, z$$ be complex numbers such that $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$$, $$\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y}=64$$, and $$\frac{x^{3}}{y+z}+\frac{y^{3}}{z+x}+\frac{z^{3}}{x+y}=488$$. If $$\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$$ where $$m, n$$ are positive integers with $$\gcd(m,n)=1$$, find $$m+n$$.

Let $$x, y$$ be real numbers such that $$xy=1$$. Let $$T$$ and $$t$$ be the largest and the smallest values of the expression $$\frac{(x+y)^{2}-(x-y)-2}{(x+y)^{2}+(x-y)-2}$$. If $$T+t$$ can be expressed in the form $$\frac{m}{n}$$ where $$m, n$$ are nonzero integers with $$\gcd(m,n)=1$$, find the value of $$m+n$$.

Let $$a,b,c$$ be reals satisfying $$3ab+2=6b$$, $$3bc+2=5c$$, and $$3ca+2=4a$$. Let $$\mathbb{Q}$$ denote the set of all rational numbers. Given that the product $$abc$$ can take two values $$\frac{r}{s}\in\mathbb{Q}$$ and $$\frac{t}{u}\in\mathbb{Q}$$, in lowest form, find $$r+s+t+u$$.

For a positive integer $$n>1$$, let $$g(n)$$ denote the largest positive proper divisor of $$n$$ and $$f(n)=n-g(n)$$. For example, $$g(10)=5$$, $$f(10)=5$$ and $$g(13)=1$$, $$f(13)=12$$. Let $$N$$ be the smallest positive integer such that $$f(f(f(N)))=97$$. Find the largest integer not exceeding $$\sqrt{N}$$.

Let $$m,n$$ be natural numbers such that $$m+3n-5=2\operatorname{lcm}(m,n)-11\gcd(m,n)$$. Find the maximum possible value of $$m+n$$.

Consider a string of $$n$$ $$1$$'s. We wish to place $$+$$ signs in between so that the sum is $$1000$$. For instance, if $$n=190$$, one may put $$+$$ signs so as to get $$11$$ ninety times and $$1$$ ten times, and get the sum $$1000$$. If $$a$$ is the number of positive integers $$n$$ for which it is possible to place $$+$$ signs so as to get the sum $$1000$$, then find the sum of the digits of $$a$$.

For an integer $$n\geq 3$$ and a permutation $$\sigma=(p_{1},p_{2},...,p_{n})$$ of $$\{1,2,...,n\}$$, we say $$p_{l}$$ is a landmark point if $$2\leq l\leq n-1$$ and $$(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$$. For example, for $$n=7$$ the permutation $$(2,7,6,4,5,1,3)$$ has four landmark points: $$p_{2}=7$$, $$p_{4}=4$$, $$p_{5}=5$$ and $$p_{6}=1$$. For a given $$n\geq 3$$, let $$L(n)$$ denote the number of permutations of $$\{1,2,...,n\}$$ with exactly one landmark point. Find the maximum $$n\geq 3$$ for which $$L(n)$$ is a perfect square.

An ant is at a vertex of a cube. Every $$10$$ minutes it moves to an adjacent vertex along an edge. If $$N$$ is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of $$N$$.

A binary sequence is a sequence in which each term is equal to $$0$$ or $$1$$. A binary sequence is called friendly if each term is adjacent to at least one term that is equal to $$1$$. For example, the sequence $$0,1,1,0,0,1,1,1$$ is friendly. Let $$F_{n}$$ denote the number of friendly binary sequences with $$n$$ terms. Find the smallest positive integer $$n\geq 2$$ such that $$F_{n}>100$$.

In a triangle $$ABC$$, the median $$AD$$ divides $$\angle BAC$$ in the ratio $$1:2$$. Extend $$AD$$ to $$E$$ such that $$EB$$ is perpendicular to $$AB$$. Given that $$BE=3$$, $$BA=4$$, find the integer nearest to $$BC^{2}$$.

Let $$N$$ be the number of ways of distributing $$52$$ identical balls into $$4$$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $$6$$. If $$N=100a+b$$, where $$a, b$$ are positive integers less than $$100$$, find $$a+b$$.