geometry problems from New Zealand Mathematical Olympiad Camp Selection Problems with aops links
collected inside aops here
NZMOC 2010-18
AB is a chord of length 6 in a circle of radius 5 and centre O. A square is inscribed in the sector OAB with two vertices on the circumference and two sides parallel to AB. Find the area of the square.
The diagonals of quadrilateral ABCD intersect in point E. Given that |AB| =|CE|, |BE| = |AD|, and \angle AED = \angle BAD, determine the ratio |BC|:|AD|.
In a convex pentagon ABCDE the areas of the triangles ABC, ABD, ACD and ADE are all equal to the same value x. What is the area of the triangle BCE?
A line drawn from the vertex A of the equilateral triangle ABC meets the side BC at D and the circumcircle of the triangle at point Q. Prove that \frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}.
Let an acute angled triangle ABC be given. Prove that the circles whose diameters are AB and AC have a point of intersection on BC.
Let a square ABCD with sides of length 1 be given. A point X on BC is at distance d from C, and a point Y on CD is at distance d from C. The extensions of: AB and DX meet at P, AD and BY meet at Q, AX and DC meet at R, and AY and BC meet at S. If points P, Q, R and S are collinear, determine d.
In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that the largest angle of the triangle is between 90^o and 135^o. If the altitudes from both B and from C are tangent to the circumcircle, then what are the angles of the triangle?
Let a point P inside a parallelogram ABCD be given such that \angle APB +\angle CPD = 180^o. Prove that AB \cdot AD = BP \cdot DP + AP \cdot CP.
From a square of side length 1, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?
Let ABCD be a quadrilateral in which every angle is smaller than 180^o. If the bisectors of angles \angle DAB and \angle DCB are parallel, prove that \angle ADC = \angle ABC
Let ABCD be a trapezoid, with AB \parallel CD (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at O. The base angles \angle DAB and \angle CBA are both acute. A point M on the line sgement OA is such that \angle BMD = 90^o, and a point N on the line segment OB is such that \angle ANC = 90^o. Prove that triangles OMN and OBA are similar.
Let C be a cube. By connecting the centres of the faces of C with lines we form an octahedron O. By connecting the centers of each face of O with lines we get a smaller cube C'. What is the ratio between the side length of C and the side length of C'?
ABCD is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center I, and a circumscribed circle with center O. Let S be the point of intersection of the diagonals of ABCD. Show that if any two of S, I and O coincide, then ABCD is a square (and hence all three coincide).
Let ABC be a triangle with \angle CAB > 45^o and \angle CBA > 45^o. Construct an isosceles right angled triangle RAB with AB as its hypotenuse and R inside ABC. Also construct isosceles right angled triangles ACQ and BCP having AC and BC respectively as their hypotenuses and lying entirely outside ABC. Show that CQRP is a parallelogram.
Let ABC be a triangle in which the length of side AB is 4 units, and that of BC is 2 units. Let D be the point on AB at distance 3 units from A. Prove that the line perpendicular to AB through D, the angle bisector of \angle ABC, and the perpendicular bisector of BC all meet at a single point.
Let ABC be an acute angled triangle. Let the altitude from C to AB meet AB at C' and have midpoint M, and let the altitude from B to AC meet AC at B' and have midpoint N. Let P be the point of intersection of AM and BB' and Q the point of intersection of AN and CC'. Prove that the point M, N, P and Q lie on a circle.
Let AB be a line segment with midpoint I. A circle, centred at I has diameter less than the length of the segment. A triangle ABC is tangent to the circle on sides AC and BC. On AC a point X is given, and on BC a point Y is given such that XY is also tangent to the circle (in particular X lies between the point of tangency of the circle with AC and C, and similarly Y lies between the point of tangency of the circle with BC and C. Prove that AX \cdot BY = AI \cdot BI.
Let ABC be an acute angled triangle. The arc between A and B of the circumcircle of ABC is reflected through the line AB, and the arc between A and C of the circumcircle of ABC is reflected over the line AC. Obviously these two reflected arcs intersect at the point A. Prove that they also intersect at another point inside the triangle ABC.
Let ABC be an acute-angled scalene triangle. Let P be a point on the extension of AB past B, and Q a point on the extension of AC past C such that BPQC is a cyclic quadrilateral. Let N be the foot of the perpendicular from A to BC. If NP = NQ then prove that N is also the centre of the circumcircle of APQ.
Consider an equilateral triangle ABC. Let P be an arbitrary point on the shorter arc AC of the circumcircle of ABC. Show that PB = PA + PC.
Altitudes AD and BE of an acute triangle ABC intersect at H. Let P \ne E be the point of tangency of the circle with radius HE centred at H with its tangent line going through point C, and let Q \ne E be the point of tangency of the circle with radius BE centred at B with its tangent line going through C. Prove that the points D, P and Q are collinear.
Let ABCD be a parallelogram with an acute angle at A. Let G be the point on the line AB, distinct from B, such that CG = CB. Let H be the point on the line BC, distinct from B, such that AB = AH. Prove that triangle DGH is isosceles.
Let ABCD be a quadrilateral. The circumcircle of the triangle ABC intersects the sides CD and DA in the points P and Q respectively, while the circumcircle of CDA intersects the sides AB and BC in the points R and S. The lines BP and BQ intersect the line RS in the points M and N respectively. Prove that the points M, N, P and Q lie on the same circle.
Let P be a point inside triangle ABC such that \angle CPA = 90^o and \angle CBP = \angle CAP. Prove that \angle P XY = 90^o, where X and Y are the midpoints of AB and AC respectively.
The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.
Let \lambda be a line and let M, N be two points on \lambda. Circles \alpha and \beta centred at A and B respectively are both tangent to \lambda at M, with A and B being on opposite sides of \lambda. Circles \gamma and \delta centred at C and D respectively are both tangent to \lambda at N, with C and D being on opposite sides of \lambda. Moreover A and C are on the same side of \lambda. Prove that if there exists a circle tangent to all circles \alpha, \beta, \gamma, \delta containing all of them in its interior, then the lines AC, BD and \lambda are either concurrent or parallel.
NZMO 2019-22
In triangle ABC, points D and E lie on the interior of segments AB and AC, respectively,such that AD = 1, DB = 2, BC = 4, CE = 2 and EA = 3. Let DE intersect BC at F. Determine the length of CF.
Let ABCDEF be a convex hexagon containing a point P in its interior such that PABC and PDEF are congruent rectangles with PA = BC = P D = EF (and AB = PC = DE = PF). Let \ell be the line through the midpoint of AF and the circumcentre of PCD. Prove that \ell passes through P.
Let X be the intersection of the diagonals AC and BD of convex quadrilateral ABCD. Let P be the intersection of lines AB and CD, and let Q be the intersection of lines PX and AD. Suppose that \angle ABX = \angle XCD = 90^o. Prove that QP is the angle bisector of \angle BQC.
Let ABCD be a square and let X be any point on side BC between B and C. Let Y be the point on line CD such that BX = YD and D is between C and Y . Prove that the midpoint of XY lies on diagonal BD.
Let \vartriangle ABC be an acute triangle with AB > AC. Let P be the foot of the altitude from C to AB and let Q be the foot of the altitude from B to AC. Let X be the intersection of PQ and BC. Let the intersection of the circumcircles of triangle \vartriangle AXC and triangle \vartriangle PQC be distinct points: C and Y . Prove that PY bisects AX.
Let \Gamma_1 and \Gamma_2 be circles internally tangent at point A, with \Gamma_1 inside \Gamma_2. Let BC be a chord of \Gamma_2 which is tangent to \Gamma_1 at point D. Prove that line AD is the angle bisector of \angle BAC
Let ABCD be a trapezium such that AB\parallel CD. Let E be the intersection of diagonals AC and BD. Suppose that AB = BE and AC = DE. Prove that the internal angle bisector of \angle BAC is perpendicular to AD.
Let ABC be an isosceles triangle with AB = AC. Point D lies on side AC such that BD is the angle bisector of \angle ABC. Point E lies on side BC between B and C such that BE = CD. Prove that DE is parallel to AB.
Let ABCD be a convex quadrilateral such that AB + BC = 2021 and AD = CD. We are also given that \angle ABC = \angle CDA = 90^o. Determine the length of the diagonal BD.
Let AB be a chord of circle \Gamma. Let O be the centre of a circle which is tangent to AB at C and internally tangent to \Gamma at P. Point C lies between A and B. Let the circumcircle of triangle POC intersect \Gamma at distinct points P and Q. Prove that \angle{AQP}=\angle{CQB}.
ABCD is a rectangle with side lengths AB = CD = 1 and BC = DA = 2. Let M be the midpoint of AD. Point P lies on the opposite side of line MB to A, such that triangle MBP is equilateral. Find the value of \angle PCB.
Let M be the midpoint of side BC of acute triangle ABC. The circle centered at M passing through A intersects the lines AB and AC again at P and Q, respectively. The tangents to this circle at P and Q meet at D. Prove that the perpendicular bisector of BC bisects segment AD.
Triangle ABC is right-angled at B and has incentre I. Points D, E and F are the points where the incircle of the triangle touches the sides BC, AC and AB respectively. Lines CI and EF intersect at point P. Lines DP and AB intersect at point Q. Prove that AQ = BF.
No comments:
Post a Comment