Canada 1969 - 2019 (CMO) 70p

with aops links in the names

1969 - 2019
Let $c$ be the length of the hypotenuse of a right angle triangle whose two other sides have lengths $a$ and $b$. Prove that $a+b\le c\sqrt{2}$. When does the equality hold?

Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, $\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}.$

Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is $\frac{2ab\cos \frac{C}{2}}{a+b}.$

Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.

Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.

Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequality $2\le a^2+b^2+c^2+d^2\le 4.$

Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.

$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. . Given $EC=1$, find the radius of the circle.

$ABCD$ is a quadrilateral with $AD=BC$. If $\angle ADC$ is greater than $\angle BCD$, prove that $AC>BD$.

A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon.
a) Prove that the sum of the lengths of these perpendiculars is constant.
b) Express this constant in terms of the radius $r$.

Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

Describe a construction of quadrilateral $ABCD$ given:
(i) the lengths of all four sides;
(ii) that $AB$ and $CD$ are parallel;
(iii) that $BC$ and $DA$ do not intersect.

Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.

If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).

Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then $\angle DBC+\angle DPC = \angle DQC.$

Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel.

$A,B,C,D$ are four "consecutive" points on the circumference of a circle and $P, Q, R, S$ are points on the circumference which are respectively the midpoints of the arcs $AB,BC,CD,DA$. Prove that $PR$ is perpendicular to $QS$.

Let $AB$ be a diameter of a circle, $C$ be any fixed point between $A$ and $B$ on this diameter, and $Q$ be a variable point on the circumference of the circle. Let $P$ be the point on the line determined by $Q$ and $C$ for which $\frac{AC}{CB}=\frac{QC}{CP}$. Describe, with proof, the locus of the point $P$.

If $A,B,C,D$ are four points in space, such that $\angle ABC= \angle BCD = \angle CDA = \angle DAB =\pi/2,$ prove that $A,B,C,D$ lie in a plane.

Let $O$ be the centre of a circle and $A$ a fixed interior point of the circle different from $O$.  Determine all points $P$ on the circumference of the circle such that the angle $OPA$ is a maximum.

A right circular cone has base radius 1 cm and slant height 3 cm is given.  $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram).  What is the minimum distance from the vertex $V$ to this path?

1978 CMO problem 4
The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$.  Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively.  Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.

1979 CMO problem 2
It is known in Euclidean geometry that the sum of the angles of a triangle is constant.  Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.

1980 CMO problem 3
Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

1980 CMO problem 5
A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property.

1981 CMO problem 2
Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle.  From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$.  Determine the maximum of the area of triangle $PQR$.

1982 CMO problem 1
In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$).  Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$.

The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$, $B'$, $C'$, and $D'$, where $AA' = k/h_a$, $BB' = k/h_b$, $CC' = k/h_c$, and $DD' = k/h_d$.  Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$, etc.  Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$.

1983 CMO problem 3
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

1984 CMO problem 4
An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.

1985 CMO problem 1
The lengths of the sides of a triangle are 6, 8 and 10 units.  Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively.  Find $AC$.

1986 CMO problem 3
A chord $ST$ of constant length slides around a semicircle with diameter $AB$. $M$ is the midpoint of $ST$ and $P$ is the foot of the perpendicular from $S$ to $AB$. Prove that $\angle SPM$ is constant for all positions of $ST$.

1987 CMO problem 3
Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$.  If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?

1988 CMO problem 2
A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within $5$ metres of the house. How much land do the garden and house together occupy?

1989 CMO problem 2
Let $ABC$ be a right angled triangle of area 1. Let $A'B'C'$ be the points obtained by reflecting $A,B,C$ respectively, in their opposite sides. Find the area of $\triangle A'B'C'.$

1990 CMO problem 3
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.

1991 CMO problem 3
Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$. Show that the midpoints of these chords lie on a circle.

1992 CMO problem 2
In the diagram, $ABCD$ is a square, with $U$ and $V$ interior points of the sides $AB$ and $CD$ respectively. Determine all the possible ways of selecting $U$ and $V$ so as to maximize the area of the quadrilateral $PUQV$.

1993 CMO problem 3
In triangle $ABC,$ the medians to the sides $\overline{AB}$ and $\overline{AC}$ are perpendicular. Prove that $\cot B+\cot C\ge \frac23.$

Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$. Suppose that the line through $PA$ cuts $\Omega$ again at $U$, and the line through $PB$ cuts $\Omega$ at $V$. Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$. Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$. Determine $\cos \angle APB$ in terms of $s,t$.

1994 CMO problem 5
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.

Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$.De termine $\angle A$.

1997 CMO problem 4
The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.

1998 CMO problem 3
Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$.  Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$.  Let $F$ be the point of intersection of the lines $BD$ and $CE$.  Show that the line $AF$ is perpendicular to the line $BC$.

Let $ABC$ be an equilateral triangle of altitude 1.  A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$.  Prove that the arc of the circle that is inside the triangle always has the same length.

2000 CMO problem 4
Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and  $AB = CB$. Prove that $AD = CD$.

2001 CMO problem 3
Let $ABC$ be a triangle with $AC > AB$.  Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$.  Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$.  Let $Z$ be the intersection point of $XY$ and $BC$.
Determine the value of $\frac{BZ}{ZC}$.

2001 CMO problem 5
Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$.
a)  Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear.
b)  Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.

Let $\Gamma$ be a circle with radius $r$.  Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$.  Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$.  Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral.  Finally, let the line $CP$ meet $\Gamma$ again at $Q$.  Prove that $PQ = r$.

2003 CMO problem 4
Prove that when three circles share the same chord $AB$, every line through $A$ different from $AB$ determines the same ratio $X Y : Y Z$, where $X$ is an arbitrary point different from $B$ on the ﬁrst circle while $Y$ and $Z$ are the points where AX intersects the other two circles (labeled so that $Y$ is between $X$ and $Z$).

2004 CMO problem 3
Let $A,B,C,D$ be four points on a circle (occurring in clockwise order), with $AB<AD$ and $BC>CD$. The bisectors of angles $BAD$ and $BCD$ meet the circle at $X$ and $Y$, respectively. Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that $BD$ must be a diameter of the circle.

2005 CMO problem 4
Let $ABC$ be a triangle with circumradius $R$, perimeter $P$ and area $K$. Determine the maximum value of: $\frac{KP}{R^3}$.

2006 CMO problem 2
Let $ABC$ be acute triangle. Inscribe a rectangle $DEFG$ in this triangle such that $D\in AB,E\in AC,F\in BC,G\in BC$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$.

2006 CMO problem 5
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $BC$ and arc $AC$ are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines $AB,AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where tangent at $D$ intersects the extended lines $AB,AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.

2007 CMO problem 5
Let the incircle of triangle $ABC$ touch sides $BC,\, CA$ and $AB$ at $D,\, E$ and $F,$ respectively. Let $\omega,\,\omega_{1},\,\omega_{2}$ and $\omega_{3}$ denote the circumcircles of triangle $ABC,\, AEF,\, BDF$ and $CDE$ respectively.
Let $\omega$ and $\omega_{1}$ intersect at $A$ and $P,\,\omega$ and $\omega_{2}$ intersect at $B$ and $Q,\,\omega$ and $\omega_{3}$ intersect at $C$ and $R.$
$a.$ Prove that $\omega_{1},\,\omega_{2}$ and $\omega_{3}$ intersect in a common point.
$b.$ Show that $PD,\, QE$ and $RF$ are concurrent.

$ABCD$ is a convex quadrilateral for which $AB$ is the longest side. Points $M$ and $N$ are located on sides $AB$ and $BC$ respectively, so that each of the segments $AN$ and $CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $MN$ bisects the diagonal $BD$.

2010 CMO problem 2
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$

2011 CMO problem 2
Let $ABCD$ be a cyclic quadrilateral with opposite sides not parallel. Let $X$ and $Y$ be the intersections of $AB,CD$ and $AD,BC$ respectively. Let the angle bisector of $\angle AXD$ intersect $AD,BC$ at $E,F$ respectively, and let the angle bisectors of $\angle AYB$ intersect $AB,CD$ at $G,H$ respectively. Prove that $EFGH$ is a parallelogram.

2012 CMO problem 3
Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.

2013 CMO problem 3
Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.

2014 CMO problem 4
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

2015 CMO problem 4
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that $BP\cdot CQ = AP\cdot AQ.$ Prove that $I$ is tangent to the circumcircle of triangle $BOC$.

2016 CMO problem 5
Let $\triangle ABC$ be an acute-angled triangle with altitudes $AD$ and $BE$ meeting at $H$. Let $M$ be the midpoint of segment $AB$, and suppose that the circumcircles of $\triangle DEM$ and $\triangle ABH$ meet at points $P$ and $Q$ with $P$ on the same side of $CH$ as $A$. Prove that the lines $ED, PH,$ and $MQ$ all pass through a single point on the circumcircle of $\triangle ABC$.

2017 CMO problem 4
Let $ABCD$ be a parallelogram. Points $P$ and $Q$ lie inside $ABCD$ such that $\bigtriangleup ABP$ and $\bigtriangleup{BCQ}$ are equilateral. Prove that the intersection of the line through $P$ perpendicular to $PD$ and the line through $Q$ perpendicular to $DQ$ lies on the altitude from $B$ in $\bigtriangleup{ABC}$.

2018 CMO problem 2
Let five points on a circle be labelled $A, B, C, D$, and $E$ in clockwise order. Assume $AE = DE$ and let $P$ be the intersection of $AC$ and $BD$. Let $Q$ be the point on the line through $A$ and $B$ such that $A$ is between $B$ and $Q$ and $AQ = DP$ Similarly, let $R$ be the point on the line through $C$ and $D$ such that $D$ is between $C$ and $R$ and $DR = AP$. Prove that $PE$ is perpendicular to $QR$.

Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is:
$(a)$ greater than 7
$(b)$ greater than 2019