IOQM GEOMETRY
Class.mathfreaks @gmail.Com
1)A triangle with perimeter 7 has integer side lengths.What is the maximum possible area of such a triangle ?
[PRMO 2012]
2)In ∆ABC , we have AC=BC=7 and AB=2. Suppose that D is a point on line AB such that B lies between A and D and CD=8. What is the length of the segment BD.
[PRMO 2012]
3)In a rectangle ABCD with AB=5 and BC=3. Points F and G are on the line segment CD so that DF=1 and GC=2. Lines AF and BG intersect at E . What is the area of ∆AEB ?
[PRMO 2012]
4)ABCD is a square and AB=1. Equilateral triangle AYB and CXD are drawn such that X and Y are inside the square . What is the length of XY.
[PRMO 2012]
5)O and I are the circumcentre and incentre of ∆ABC respectively. Suppose O lies in the interior of ∆ABC and I lies on the circle passing through B, O and C. What is the magnitude of angle BAC in degrees?
[PRMO 2012]
6)PS is a line segment of length 4 and O is the midpoint of PS. A semi-circular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. What is the area of the region QXROQ bounded by the two semi circular arcs?
[PRMO 2012]
7)Three points X,Y,Z are on a striaght line such that XY=10 and XZ=3. What is the product of all possible values of YZ?
[PRMO 2013]
8)Let AD and BC be the parallel sides of a trapezium ABCD. Let P and Q be the midpoints of the diagonals AC and BD. If AD=16 and BC=20, what is the length of PQ?
[PRMO 2013]
9)In a triangle ABC, let H,I and O be the orthocentre, incentre and circumcentre, respectively. If the points B,H,I,C lie on a circle, what is the magnitude of ∠BOC in degrees?
[PRMO 2013]
10)Let ABC be an equilateral triangle. Let P and S be points on AB and AC, respectively, and let Q and R be points on BC such that PQRS is a rectangle. If PQ=3√PS and the area of PQRS is 28√3, what is the length of PC?
[PRMO 2013]
11)Let A₁,B₁,C₁,D₁ be the midpoints of the sides of a convex quadrilateral ABCD and let A₂,B₂, C₂,D₂ be the midpoints of the sides of the quadrilateral A₁B₁C₁D₁. If A₂B₂C₂D₂ is a rectangle with sides 4 and 6, then what is the product of the lengths of the diagonals of ABCD?
[PRMO 2013]
13)Let S be a circle with centre O. A chord AB, not a diameter, divides S into two regions R₁ and R₂ such that O belongs to R₂. Let S₁ be a circle with centre in R₁, touching AB at X and S internally. Let S₂ be a circle with centre in R₂, touching AB at Y, the circle S internally and passing through the centre of S. The point X lies on the diameter passing through the centre of S₂ and ∠YXO=30∘. If the radius of S₂ is 100 then what is the radius of S₁?
[PRMO 2013]
14)In a triangle ABC with ∠BCA=90∘, the perpendicular bisector of AB intersects segments AB and AC at X and Y, respectively. If the ratio of the area of quadrilateral BXYC to the area of triangle ABC is 13: 18 and BC=12 then what is the length of AC?
[PRMO 2013]
15)Let ABCD be a convex quadrilateral with perpendicular diagonals. If AB=20,BC=70 and CD=90, then what is the value of DA?
[PRMO 2014]
16)In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?
[PRMO 2014]
17)In a triangle ABC,X and Y are points on the segments AB and AC, respectively, such that AX:XB=1:2 and AY:YC=2:1. If the area of triangle AXY is 10 then what is the area of triangle ABC˙?
[PRMO 2014]
18)Let ABCD be a convex quadrilateral with ∠DAB=∠BDC=90∘. Let the incircles of triangles ABD and BCD touch BD at P and Q, respectively, with P lying in between B and Q. If AD=999 and PQ=200 then what is the sum of the radii of the incircles of triangles ABD and BDC?
[PRMO 2014]
19)Let XOY be a triangle with ∠XOY=90∘. Let M and N be the midpoints of legs OX and OY, respectively. Suppose that XN=19 and YM=22. What is XY?
[PRMO 2014]
20)In a triangle ABC, let I denote the incenter. Let the lines AI,BI and CI intersect the incircle at P,Q and R, respectively. If ∠BAC=40∘, what is the value of ∠QPR in degrees?
[PRMO 2014]
21)The figure below shows a broken piece of a circular plate made of glass.
C is the midpoint of AB, and D is the midpoint of arc AB. Given that AB = 24 cm and CD = 6 cm, what is the radius of the plate in centimeters? (The figure is not drawn to scale.)
[PRMO 2015]
22)A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
[PRMO 2015]
23)What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ?
[PRMO 2015]
24)In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that ∠BPC = 90◦ . If r1,r2,r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value of r1+r2+r3?
[PRMO 2015]
25)In acute-angled triangle ABC, let D be the foot of the altitude from A, and E be the midpoint of BC. Let F be the midpoint of AC. Suppose ∠BAE=40◦.If∠DAE=∠DFE, what is the magnitude of ∠ADF in degrees?
[PRMO 2015]
26)The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume PY>PX. Let PY intersect ω at Z. If YZ=2PZ, what is the magnitude of ∠PYX in degrees?
[PRMO 2015]
27). Suppose in ∆ABC, ∠ABC =90◦, AB = BC, and AC =√3−1. Suppose there exists a point P₀ in the plane of ∆ABC such that AP₀+ BP₀+ CP₀≤ AP + BP + CP for all points P in the plane of ∆ABC. Find AP₀ + BP₀ + CP₀.
[PRMO 2015 WEST BENGAL REGION]
28)Let ∆ABC be an equilateral triangle with each side 2√3. Let P be a point outside the triangle such that the points A and P lie in the opposite sides of the straight line BC. Let P D, P E, P F be the perpendiculars dropped on the sides BC, AC and AB respectively where D, foot of the perpendicular, lies inside the line segment BC. Let PD = 2. What is the value of PE+ PF?
[PRMO 2015 WEST BENGAL REGION]
29)In trapezium PQRS, QR||PS. Let QR = 1001, P S = 2015. Also, let ∠P = 37◦ and ∠S = 53◦. Finally, let X and Y be the midpoints of QR and P S, respectively. Find the length of XY .
[PRMO 2015 WEST BENGAL REGION]
30)A square PQRS length of its side equal to 3 +√5. Let M be the mid-point of the side RS. Also, let C₁ be the in-circle of ∆PMS and C₂ be the circle that touches the sides PQ,QR and PM. Find the radius of the circle C₂.
[PRMO 2015 WEST BENGAL REGION]
31)Let AD be an altitude in a right triangle ABC with ∠A = 90◦ and D on BC. Suppose that the radii of the in-circles of the triangles ABD and ACD are 33 and 56 respectively. Let r be the radius of the in-circle of triangle ABC. Find the value of 3(r+7).
[ PRMO 2016]
32)In triangle ABC right angled at vertex B, a point O is chosen on the side BC such that the circle γ centered at O of radius OB touches the side AC. Let AB = 63 and BC = 16, and the radius of γ be of the form m/n where m, n are relatively prime positive integers. Find the value of m+n .
[PRMO 2016]
33)The hexagon OLYMPI has a reflex angle at O and convex at every other vertex. Suppose that LP=3√2 units and the condition ∠O=10∠L=2∠Y=5∠M=2∠P=10∠I holds. Find the area (in sq units) of the hexagon.
[PRMO 2016]
34)Points G and O denote the centroid and the circumcenter of the triangle ABC. Suppose that ∠AGO = 90◦ and AB = 17,AC = 19. Find the value of BC² .
[PRMO 2016]
35)In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC =8√(3) .Find AB.
[PRMO 2016]
36)Consider a right-angled triangle ABC with∠C = 90⁰. Suppose that the hypotenuse AB is divided into four equal parts by the points D, E, F, such that AD = DE = EF = F B. If CD² + CE²+ CF² = 350, find the length of AB.
[PRMO WEST BENGAL REGION 2016]
37) Consider a triangle ABC with AB = 13, BC = 14, CA = 15. A line perpendicular to BC divides the interior of ∆ABC into two regions of equal area.Suppose that the aforesaid perpendicular cuts BC at D, and cuts ∆ABC again at E. If L is the length of the line segment DE, find L2.
[PRMO WEST BENGAL REGION 2016]
38) Suppose a circle C of radius √2 touches the Y -axis at the origin (0, 0). A ray of light L, parallel to the X-axis, reflects on a point P on the circumference of C, and after reflection, the reflected ray L' becomes parallel to the Y -axis.Find the distance between the ray L and the X-axis.
[PRMO WEST BENGAL REGION 2016]
39)Suppose the altitudes of a triangle are 10, 12, and 15. What is its semi-perimeter?
[PRMO 2017]
40)Let P be an interior point of a triangle ABC whose side lengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length.
[PRMO 2017]
41)Let ABCD be a rectangle and let E and F be points on CD and BC respectively such that area(ADE) = 16, area(CEF)=9 and area(ABF) = 25. What is the area of triangle AEF?
[PRMO 2017]
42)Let AB and CD be two parallel chords in a circle with radius 5 such that the center O lies between these chords. Suppose AB = 6, CD = 8. Suppose further that the area of the part of the circle lying between the chords AB and CD is (mπ+n)/k, where m, n, k are positive integers with gcd(m,n,k)=1. What is the value of m+n+k?
[PRMO 2017]
43)Let Ω₁ be a circle with center O and let AB be a diameter of Ω₁ . Let P be a point on the segment OB different from O. Suppose another circle Ω₂ with center P lies in the interior of Ω₁. Tangents are drawn from A and B to the circle Ω₂ intersecting Ω₁ again at A₁ and B₁ respectively such that A₁ and B₁ are on the opposite sides of AB. Given that A₁B=5,AB₁=15 and OP=10, find the radius of Ω₁
[PRMO 2017]
44)Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at a time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.
[PRMO 2017]
45)In a quadrilateral ABC , it is given that AB=AD=13,BC=CD=20,BD=24. If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?
[PRMO 2018]
46)Let ABCD be a trapezium in which AB||CD and AD⊥AB. Suppose ABCD has an in-circle which touches AB at Q and CD at P. Given that PC = 36 and QB = 49, find PQ.
[PRMO 2018]
47)A point P in the interior of a regular hexagon is at distances 8,8,16 units from three consecutive vertices of the hexagon, respectively. If r is radius of the circumscribed circle of the hexagon, what is the integer closest to r?
[PRMO 2018]
48)Let AB be a chord of a circle with center O. Let C be a point on the circle such that ∠ABC=30⁰ and O lies inside the triangle ABC. Let D be a point on AB such that ∠DCO=∠OCB=20⁰. Find the measure of ∠CDO in degrees.
[PRMO 2018]
49)In a triangle ABC, the median from B to CA is perpendicular to the median from C to AB. If the median from A to BC is 30, determine (BC²+CA²+AB²)/100.
[PRMO 2018]
50)In a triangle ABC, right-angled at A, the altitude through A and the internal bisector of ∠A have lengths 3 and 4, respectively. Find the length of the median through A.
[PRMO 2018]
51)Let ABC be an acute-angled triangle and let H be its orthocentre. Let G₁,G₂ and G₃ be the centroids of the triangles HBC, HCA and HAB, respectively. If the area of triangle G₁G₂G₃ is 7 units, what is the area of triangle ABC?
[PRMO 2018]
52)Let D be an interior point of the side BC of a triangle ABC. Let I₁ and I₂ be the incentres of triangles ABD and ACD respectively. Let AI₁ and AI₂ meet BC in E and F respectively. If ∠BI₁E=600, what is the measure of ∠CI₂F in degrees?
[PRMO 2018]
53)From a square with sides of length 5, triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer?
[PRMO 2019 (11-08-2019)]
54)Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A₂, B₁ and C₁ respectively and the internal bisectors of angles A₁, B₁ and C₁ of the triangle A₁B₁C₁ intersect at A₂, B₂ and C₂, respectively. If the smallest angle of triangle ABC is 40°, what is the magnitude of the smallest angle of triangle A₂B₂C₂ in degrees?
[PRMO 2019 (11-08-2019)]
55)Let AB be a diameter of a circle and let C be a point on the segment AB such that AC : CB = 6 : 7. Let D be a point on the circle such that DC is perpendicular to AB. Let DE be the diameter through D. If [XYZ] denotes the area of the triangle XYZ. Find [ABD]/[CDE] to the nearest integer.
[PRMO 2019 (11-08-2019)]
56) Let ABCD be a convex cyclic quadrilateral. Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least. If {PA, PB, PC, PD} = {3, 4, 6, 8}
[PRMO 2019 (11-08-2019)]
57)A village has a circular wall around it, and the wall has four gates pointing north, south east and west. A tree stands outside the village, 16m north of the north gate, and it can be just seen appearing on the horizon from a point 48 m east of the south gate. What is the diameter in meters, of the wall that surrounds the village ?
[PRMO 2019 (11-08-2019)]
58)Let ABC be a triangle with sides 51, 52, 53. Let Ω denote the incircle of ABC. Draw tangents to which are parallel to the sides of ABC. Let r₁,r₂,r₃ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed r₁ + r₂ + r₃.
[PRMO 2019 (11-08-2019)]
59)In a triangle ABC, the median AD (with D on BC) and the angle bisector BE (with E on AC) are perpendicular to each other. If AD = 7 and BE = 9, find the integer nearest to the area of triangle ABC.
[PRMO 2019 (11-08-2019)]
60)Let ABC be a triangle such that AB = AC. Suppose the tangent to the circumcircle of ABC at B is perpendicular of AC. Find ABC measured in degrees.
[PRMO 2019 (25-08-2019)]
61)The centre of the circle passing through the midpoints of the sides of an isosceles triangle ABC lies on the circumcircle of triangle ABC. If the larger angle of triangle ABC is α⁰ and the smaller one β⁰ then what is the value of α-β ?
[PRMO 2019 (25-08-2019)]
62)If 15 and 9 are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?
[PRMO 2019 (25-08-2019)]
63)In parallelogram ABCD, AC = 10 and BD = 28. The points K and L in the plane of ABCD move in such a way that AK = BD and BL = AC. Let M and N be the midpoints of CK and DL, respectively. What is the maximum value of cot²(BMD/2) + tan²(ANC/2) ?
[PRMO 2019 (25-08-2019)]
64)Let ABC be an isosceles triangle with AB = BC. A trisector of B meets AC at D. If AB, AC and BD are integers and AB – BD = 3, find AC.
[PRMO 2019 (25-08-2019)]
65)A friction-less board has the shape of an equilateral triangle of side length 1 meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex A towards the side BC. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equal the angle of reflection. The distance traveled by the ball in meters is of the form N, where N is an integer. What is the value of N ?
[PRMO 2019 (25-08-2019)]
66)A conical glass is in the form of a right circular cone. The slant height is 21 and the radius of the top rim of the glass is 14. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. (see the figure ). If d the shortest distance it should crawl to reach the honey drop, what is the integer part of d ? (Ignore the thickness of the glass)
[PRMO 2019 (25-08-2019)]
67)In a triangle ABC, it is known that A = 100° and AB = AC. The internal angle bisector BD has length 20 units. Find the length of BC to the nearest integer, given that sin 10° =0.17(approx)
[PRMO 2019 (25-08-2019)]
68)Let ABCD be a trapezium in which AB k CD and AB = 3CD. Let E be the midpoint of the diagonal BD. If [ABCD] = n×[CDE], what is the value of n? (Here [Γ] denotes the area of the geometrical figure Γ.)
[IOQM 2020-21]
69)Let ABCD be a rectangle in which AB + BC + CD = 20 and AE = 9 where E is the mid-point of the side BC. Find the area of the rectangle.
[IOQM 2020-21]
70)Let ABC be a triangle with AB = AC. Let D be a point on the segment BC such that BD = 48 ¹/₆₁ and DC = 61. Let E be a point on AD such that CE is perpendicular to AD and DE = 11. Find AE.
[IOQM 2020-21]
71)Let ABC 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 k AC and DN k BC. If (MN)²=p/q where p and q are relatively prime positive integers then what is the sum of the digits of |p − q|?
[IOQM 2020-21]
72)The sides x and y of a scalene triangle satisfy x +2∆/x= y +2∆/y, where ∆ is the area of the triangle. If x = 60, y = 63, what is the length of the largest side of the triangle?
[IOQM 2020-21]
73)Let ABCD be a parallelogram . Let E and F be midpoints of AB and BC respectively. The lines EC and F D intersect in P and form four triangles APB, BPC,CPD and DPA. If the area of the parallelogram is 100 sq. units, what is the maximum area in sq. units of a triangle among these four triangles?
[IOQM 2020-21]
74)In triangle ABC, let P and R be the feet of the perpendiculars from A onto the external and internal bisectors of ∠ABC, respectively; and let Q and S be the feet of the perpendiculars from A onto the internal and external bisectors of ∠ACB,respectively. If P Q = 7, QR = 6 and RS = 8, what is the area of triangle ABC?
[IOQM 2020-21]
75) The incircle Γ of a scalene triangle ABC touches BC at D, CA at E and AB at F.Let rA be the radius of the circle inside ABC which is tangent to Γ and the sides AB and AC. Define rB and rC similarly. If rA = 16, rB = 25 and rC = 36, determine the radius of Γ.
[IOQM 2020-21]
76)A light source at the point (0, 16) in the coordinate plane casts light in all directions.A disc (a circle along with its interior) of radius 2 with center at (6, 10) casts a shadow on the X axis. The length of the shadow can be written in the form m√n where m, n are positive integers and n is square-free. Find m + n.
[IOQM 2020-21]
77)If ABCD is a rectangle and P is a point inside it such that AP = 33, BP = 16, DP =63. Find CP.
[IOQM KV 2020-21]
78)Let ABC be an isoceles triangle with AB = AC and incentre I. If AI = 3 and the distance from I to BC is 2 , what is the square of the length of BC?
[IOQM KV 2020-21]
79)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 insideABCD , is drawn to touch this circle externally and such that the circle also touches AB and AD. If r = m + n√k ,where m, n are integers and k is a prime number , find the value of √(m+n/k²)
[IOQM KV 2020-21]
80)Two sides of regular polygon when extended, meet at angle of 28⁰. What is the smallest possible value of n?
[IOQM KV 2020-21]
81) Let D, E, F be points on the sides BC, CA, AB of a triangle ABC , respectively.Suppose AD, BE, CF are concurrent at point 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.
[IOQM KV 2020-21]
82)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 l. Then find l²
[IOQM KV 2020-21]
83)Let ABC be a triangle with ∠BAC = 90◦ and D be the point on the side BC such that AD ⊥ BC. Let r,r₁, and r₂ be the inradii of triangles ABC, ABD, and ACD,respectively. If r,r₁, and r₂ are positive integers and one of them is 5 , find the largest possible value of r + r₁ + r₂.
[IOQM KV 2020-21]
84)Two circles, S₁ and S₂, of radii 6 units and 3 units respectively, are tangent to each other, externally. Let AC and BD be their direct common l tangents with A and B on S₁ and C and D on S₂. Find the area of quadrilateral ABDC to the nearest integer.
[IOQM KV 2020-21]
85)Let ABC be an acute-angled triangle and P be a point in its interior. Let PA, PB, and PC be the images of P under reflection in the sides BC, CA, and AB, respectively. If P is the orthocentre of the triangle and if the largest angle of the triangle that can be formed by the line segments PA, PB, and PC is x⁰. determine the valueof x.
[IOQM KV 2020-21]
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