geometry problems from Nordic Mathematical Contest (NMC)

with aops links in the names

with aops links in the names

NMC Geometry Problems 1987-2017 EN in pdf with aops links

NMC all 1998-2019 EN in pdf with solutions

NMC all 1998-2019 EN in pdf with solutions

NMC all 1987-2011 EN in pdf with solutions

by Matti Lehtinen, from latex

by Matti Lehtinen, from latex

1987 - 2019

Let ABCD be a parallelogram in
the plane. We draw two circles of radius R, one through the points
A and B, the other through B and C. Let E be the other point of intersection of the circles. We assume that E is not a vertex of the parallelogram. Show that the circle passing
through A, D, and E also has radius R.

Two concentric spheres have radii r and R, r < R. We try to select
points A, B and C on the surface of the larger sphere such that all sides of the triangle ABC would be tangent to the surface of the smaller sphere. Show that the
points can be selected if and only if R ≤ 2r.

Three sides of a tetrahedron are right-angled
triangles having the right angle at their common vertex. The areas of these
sides are A, B, and C. Find the total surfacearea of the tetrahedron.

Let ABC be a triangle and let P be an interior point of ABC. We assume that a line

*l*, which passes through P, but not through A, intersects AB and AC (or their extensions over B or C) at Q and R, respectively. Find*l*such that the perimeter of the triangle AQR is as small as possible.
In the trapezium ABCD the sides AB and CD are parallel, and E is a fixed point on the side AB. Determine the point F on the side CD so that the area of the intersection of the
triangles ABF and CDE is as large as possible.

Prove that among all triangles with inradius 1,
the equilateral one has the smallest perimeter.

A hexagon is inscribed in a circle of radius r. Two of the sides of the hexagon have length 1, two have length 2 and
two have length 3. Show that r
satisfies the equation 2r

^{3}− 7r − 3 = 0.
Let O be an interior point in
the equilateral triangle ABC, of side length a.
The lines AO, BO, and CO intersect the sides of the triangle in the points A

_{1}, B_{1}, and C_{1}. Show that |OA_{1}| + |OB_{1}| + |OC_{1}| < a.
A piece of paper is the square ABCD. We fold it by placing the vertex D on the point D΄
of the side BC. We assume that AD moves on the segment A΄D΄ and that A΄D΄
intersects AB at E. Prove that the perimeter of the triangle EBD΄ is one half of the perimeter of the square.

Let AB be a diameter of a
circle with centre O. We choose a point C on the circumference of the circle such that OC and AB are perpendicular to each other. Let P be an arbitrary point on the (smaller) arc BC and let the lines CP and AB meet at Q. We choose R on AP so that RQ and AB are perpendicular to each other. Show that |BQ| = |QR|.

Show that there exist infinitely many mutually
non-congruent triangles T, satisfying

(i) The side lengths of T are consecutive integers.

(ii) The area of T is an integer.

The circle whose diameter is the altitude
dropped from the vertex A of the triangle ABC intersects the sides AB
and AC at D and E, respectively (A ≠ D, A ≠ E). Show that the
circumcentre of ABC lies on the altitude dropped from the vertex A of the triangle ADE, or on its extension.

Let ABCD be a convex
quadrilateral. We assume that there exists a point P inside the quadrilateral such that the areas of the triangles ABP, BCP, CDP, and DAP are equal. Show that at least one of the diagonals of the quadrilateral
bisects the other diagonal.

Let A, B, C, and D be four different points
in the plane. Three of the line segments AB, AC, AD, BC, BD, and CD have length a. The other three have
length b, where b > a. Determine all possible
values of the quotient b / a.

Let C

_{1}and C_{2}be two circles intersecting at A and B. Let S and T be the centers of C_{1}and C_{2}, respectively. Let P be a point on the segment AB such that |AP| ≠ |BP| and P ≠ A, P ≠B. We draw a line perpendicular to SP through P and denote by C and D the points at which this line intersects C_{1}. We likewise draw a line perpendicular to TP through P and denote by E and F the points at which this line intersects C_{2}. Show that C, D, E, and F are the vertices of a rectangle.
Consider 7-gons inscribed in a circle such that
all sides of the 7-gon are of different length. Determine the maximal number of
120°
angles in this kind of a
7-gon.

In the triangle ABC,
the bisector of angle B meets AC at D and the bisector of angle C meets AB at E. The bisectors meet each other at O. Furthermore, OD = OE. Prove that either ABC is isosceles or ∠BAC = 60°.

NMC 2001.4

Let ABCDEF be a convex hexagon, in
which each of the diagonals AD, BE, and CF divides the hexagon into two quadrilaterals of equal area. Show that AD, BE, and CF are concurrent.

The trapezium ABCD, where AB and CD are parallel and AD < CD,
is inscribed in the circle c. Let DP be a chord of the circle, parallel to AC. Assume that the tangent to c at D meets the line AB at E and that PB and DC meet at Q. Show that EQ = AC.

The point D inside the equilateral triangle _ABC satisfies ∠ADC = 150°. Prove that a triangle
with side lengths |AD|, |BD|, |CD| is necessarily a right-angled triangle.

Let a, b, and c be the side lengths of a triangle and let R be its circumradius. Show that 1/ ab + 1/ bc + 1/ ca ≥ 1/ R

^{2}.
The circle C

_{1}is inside the circle C_{2}, and the circles touch each other at A. A line through A intersects C_{1}also at B and C_{2}also at C. The tangent to C_{1}at B intersects C_{2}at D and E. The tangents of C_{1}passing through C touch C_{1}at F and G. Prove that D, E, F, and G are concyclic.
Let B and C be points on two fixed rays emanating from a point A such that AB + AC is constant. Prove that
there exists a point D ≠ A such that the circumcircles of the triangles ABC pass through D for every choice of B and C.

A triangle, a line and
three rectangles, with one side parallel to the given line, are given in such a
way that the rectangles completely cover the sides of the triangle. Prove that
the rectangles must completely cover the interior of the triangle.

A line through a point A intersects a circle in two points, B and C, in such a way that B lies between A and C. From the point A draw the two tangents to
the circle, meeting the circle at points S and T. Let P be the intersection of the lines ST and AC. Show that AP
/ PC = 2 AB / BC.

Let ABC be a triangle and let D and E be points on BC
and CA, respectively, such that AD and BE are angle bisectors of ABC. Let F and G be points on the circumcircle of ABC such that AF and DE are parallel and FG and BC are parallel. Show that AG/BG = (AC + BC)/(AB +
CB).

A point P is chosen in an arbitrary triangle. Three lines
are drawn through P
which are parallel to
the sides of the triangle. The lines divide the triangle into three smaller triangles
and three parallelograms. Let f be the ratio between the
total area of the three smaller triangles and the area of the given triangle.
Show that f ≥ 1/3 and determine those points P for which f
= 1 /3 .

Three circles ΓA, ΓB and ΓC share a common point of intersection O. The other common of ΓA and ΓB is C, that of ΓA and ΓC is B and that of ΓC and ΓB is A. The line AO
intersects the circle ΓC in the poin X ≠ O. Similarly, the line BO intersects the circle ΓB in the point Y≠ O, and the line CO intersects the circle ΓC in the point Z ≠ O. Show that

|AY ||BZ||CX| / |AZ||BX||CY | = 1.

In a triangle ABC assume AB = AC, and let D and E be points on the extension of segment BA beyond A and on the segment BC, respectively, such that the lines CD and AE are parallel. Prove that CD ≥ 4h CE /BC , where h is the height from A in triangle ABC. When does equality hold?

Given a triangle ABC, let P lie on the circumcircle of
the triangle and be the midpoint of the arc BC which does not contain A. Draw a
straight line

*l*through P so that*l*is parallel to AB. Denote by*k*the circle which passes through B, and is tangent to*l*at the point P. Let Q be the second point of intersection of*k*and the line AB (if there is no second point of intersection, choose Q = B). Prove that AQ = AC.
Let ABC be
an acute angled triangle, and H a
point in its interior. Let the reflections of H through the sides AB and
AC be called Hc and Hb, respectively, and let the reflections of H through the midpoints of these same
sides be called H΄c and H΄b, respectively. Show that the four points Hb, H΄b , Hc, and H΄c are
concyclic if and only if at least two of them coincide or H lies on the altitude from A in triangle ABC.

Given an equilateral triangle, find all points inside
the triangle such that the distance from the point to one of the sides is equal
to the geometric mean of the distances from the point to the other two sides of
the triangle.

Let ABC be a triangle and Γ the circle with
diameter AB. The bisectors of ∠BAC
and ∠ABC
intersect Γ (also) at D and E, respectively. The incircle of ABC meets BC and
AC at F and G,
respectively. Prove that D, E, F and G are
collinear.

Let ABCD be a cyclic quadrilateral satisfying AB = AD
and AB + BC = CD. Determine ∠CDA.

Let M and
N be the midpoints of the sides
AC and AB, respectively, of an acute triangle ABC, AB ≠ AC.
Let ω

_{B}be the circle centered at M passing through B, and let ω_{C}be the circle centered at N passing through C. Let the point D be such that ABCD is an isosceles trapezoid with AD parallel to BC. Assume that ω_{B}and ω_{C }intersect in two distinct points P and Q. Show that D lies on the line PQ.
Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incentre of triangle $ABC$, and let $H$ be the orthocentre of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.

The quadrilateral $ABCD$ satisfies $\angle ACD = 2\angle CAB, \angle ACB = 2\angle CAD $ and $CB = CD.$ Show that $$\angle CAB=\angle CAD.$$

source: www.georgmohr.dk/nmcperm/

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