IOQM Previous Papers march 6 2021-22

Three parallel lines $$L_1,L_2,L_3$$ are drawn in the plane such that the perpendicular distance between $$L_1$$ and $$L_2$$ is $$3$$ and the perpendicular distance between $$L_2$$ and $$L_3$$ is also $$3$$. A square $$ABCD$$ is constructed such that $$A$$ lies on $$L_1$$, $$B$$ lies on $$L_3$$ and $$C$$ lies on $$L_2$$. Find the area of the square.

Ria writes down the numbers $$1,2,\ldots,101$$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers written in red. How many numbers did Ria write with red pen?

Consider the set $$\mathcal{T}$$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $$\triangle\in\mathcal{T}$$ be the triangle with the least perimeter. If $$a^\circ$$ is the largest angle of $$\triangle$$ and if $$L$$ is its perimeter, determine the value of $$\frac{a}{L}$$.

Consider the set of all $$6$$-digit numbers consisting of only $$3$$ digits, $$a,b,c$$, where $$a,b,c$$ are distinct. Suppose the sum of all these numbers is $$593999406$$. What is the largest remainder when the three-digit number $$abc$$ is divided by $$100$$?

In parallelogram $$ABCD$$ the longer side is twice the shorter side. Let $$XYZW$$ be the quadrilateral formed by the internal bisectors of the angles of $$ABCD$$. If the area of $$XYZW$$ is $$10$$, find the area of $$ABCD$$.

Let $$x,y,z$$ be positive real numbers such that $$x^2+y^2=49$$, $$y^2+yz+z^2=36$$ and $$x^2+\sqrt{3}xz+z^2=25$$. If the value of $$2xy+\sqrt{3}yz+xz$$ can be written as $$p\sqrt{q}$$ where $$p,q$$ are integers and $$q$$ is not divisible by the square of any prime, find $$p+q$$.

Find the number of maps $$f:\{1,2,3\}\rightarrow\{1,2,3,4,5\}$$ such that $$f(i)\leq f(j)$$ whenever $$i<j$$.

For any real number $$t$$, let $$\lfloor t\rfloor$$ denote the largest integer $$\leq t$$. Suppose that $$N$$ is the greatest integer such that $$\left\lfloor\sqrt{\left\lfloor\sqrt{\left\lfloor\sqrt{N}\right\rfloor}\right\rfloor}\right\rfloor=4$$. Find the sum of digits of $$N$$.

Let $$P_0=(3,1)$$ and define $$P_{n+1}=(x_n,y_n)$$ for $$n\geq0$$ by $$x_{n+1}=-\frac{3x_n-y_n}{2}$$, $$y_{n+1}=-\frac{x_n+y_n}{2}$$. Find the area of the quadrilateral formed by the points $$P_{96},P_{97},P_{98},P_{99}$$.

Suppose that $$P$$ is the polynomial of least degree with integer coefficients such that $$P(\sqrt{7}+\sqrt{5})=2(\sqrt{7}-\sqrt{5})$$. Find $$P(2)$$.

In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

A $$12 × 12$$ board is divided into $$144$$ unit squares by drawing lines parallel to the sides. Two
rooks placed on two unit squares are said to be non attacking if they are not in the same
column or same row. Find the least number $$N$$ such that if $$N$$ rooks are placed on the unit
squares, one rook per square, we can always find $$7$$ rooks such that no two are attacking
each other.