Indian IOQM 2021-22 22p

 geometry problems from Indian Olympiad Qualifier in Mathematics (IOQM) with aops links in the names

a replacement for both PRMO and RMO in India for 2020-21

2022 IOQM aops collection

2020-2021

phase 1

2021 IOQM p1

Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times  [CDE]$, what is the value of $n$?

(Here $[t]$ denotes the area of the geometrical figure$t$.)

2021 IOQM p4

Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.

2021 IOQM p7

Let $\triangle ABC$ be a triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD= 48 \frac{1}{61}$ and $DC=61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE=11$. Find $AE$.

2021 IOQM p9

Let A$BC$ be a triangle with $AB = 5, AC = 4, BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM\parallel AC$ and $DN \parallel BC$. If $(MN)^2 =\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?

2021 IOQM p12

Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$, how many chords can be drawn before returning to the starting point ?

2021 IOQM p16

The sides $x$ and $y$ of a scalene triangle satisfy $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$ , where $\Delta$ is the area of the triangle. If $x = 60, y = 63$, what is the length of the largest side of the triangle?

2021 IOQM p19 

Let $ABCD$ be a parallelogram. Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect at $P$ and form four triangles $APB, BPC, CPD, DPA$. If the area of the parallelogram is $100$, what is the maximum area of a triangles among these four triangles?

2021 IOQM p22

In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7, QR = 6$ and $RS = 8$, what is the area of triangle $ABC$?

2021 IOQM p23

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

phase 2

2021 KVS IOQM p2

If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

2021 KVS IOQM p4

Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

2021 KVS IOQM p6

Let $ABCD$ be a square with side length $100$. A circle with centre $C$ and radius $CD$ is drawn. Another circle of radius $r$, lying inside $ABCD$, is drawn to touch this circle externally and such that the circle also touches $AB$ and $AD$. If $r=m+n\sqrt{k}$, where $m,n$ are integers and $k$ is a prime number, find the value of $\frac{m+n}k$.

2021 KVS IOQM p13

The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter

2021 KVS IOQM p14

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

2021 KVS IOQM p18

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose $AD, BE,CF$ are concurrent at $P$. If $PF/PC =2/3, PE/PB = 2/7$ and $PD/PA = m/n$, where $m, n$ are positive integers with $gcd(m, n) = 1$, find $m + n$.

2021 KVS IOQM p19

A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then determine $\ell^2$.

2021 KVS IOQM p22

Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.

2021 KVS IOQM p24

Two circles $S_1$ and $S_2$, of radii $6$ units and $3$ units respectively, are tangent to each other, externally. Let $AC$ and $BD$ be their direct common tangents with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest Integer.

2021 KVS IOQM p27

Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$PA, PB$. and $PC$ is $x^o$, determine the value of $x$.

2022 IOQM p1

Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ 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.

2022 IOQM p3

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 least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

2022 IOQM p5

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 $ABCD$