### Canadian MO Qual. Repêchage 2009-18 (CMOQR) 20p

with aops links in the names

2009 - 2018

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

2009 CMOQR p6
Triangle $ABC$ is right-angled at $C$. $AQ$ is drawn parallel to $BC$ with $Q$ and $B$ on opposite sides of $AC$ so that when $BQ$ is drawn, intersecting $AC$ at $P$, we have $PQ = 2AB$. Prove that $\angle ABC = 3\angle PBC$.

A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?

2010 CMOQR p2
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

2010 CMOQR p8
Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.

2011 CMOQR p1
In the diagram, the circle has radius $\sqrt 7$ and and centre $O.$ Points $A, B$ and $C$ are on the circle. If $\angle BOC=120^\circ$ and $AC = AB + 1,$ determine the length of $AB.$

2011 CMOQR p6
In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD.$ The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ,$ for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$

2012 CMOQR p4
Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*}
S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\
S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\
S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\
S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\
S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\

S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.

2012 CMOQR p8
Suppose circles $\mathit{W}_1$ and $\mathit{W}2$, with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$. Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$. Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$. Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$. Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$, respectively.

Show that $\angle\mathit{I}_1\mathit{MI}_2=\angle\mathit{O}_1\mathit{MO}_2.$

2013 CMOQR p2
In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$.

2013 CMOQR p8
Let $\triangle ABC$ be an acute-angled triangle with orthocentre $H$ and circumcentre $O$. Let $R$ be the radius of the circumcircle. \begin{align*}
\text{Let }\mathit{A'}\text{ be the point on }\mathit{AO}\text{ (extended if necessary) for which }\mathit{HA'}\perp\mathit{AO}. \\
\text{Let }\mathit{B'}\text{ be the point on }\mathit{BO}\text{ (extended if necessary) for which }\mathit{HB'}\perp\mathit{BO}. \\

\text{Let }\mathit{C'}\text{ be the point on }\mathit{CO}\text{ (extended if necessary) for which }\mathit{HC'}\perp\mathit{CO}.\end{align*} Prove that $HA'+HB'+HC'<2R$

2014 CMOQR p4
In $\triangle ABC$, the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$. A laser beam exits the point $A_1$, hits side $AC$ at point $B_1$, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$, then side $BC$ at point $A_2$, then side $AC$ again at point $B_2$, then side $AB$ again at point $C_2$, then side $BC$ again at point $A_3$, and finally, side $AC$ again at point $B_3$.
(a) Prove that $\angle B_3A_3C = \angle B_1A_1C$.
(b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$.

2014 CMOQR p6
Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and  $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.

2015 CMOQR p4
Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.

2015 CMOQR p6
Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$.

2016 CMOQR p2
Let $P = (7, 1)$ and let $O = (0, 0)$.
(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$.
(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.

2016 CMOQR p5
Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.

2017 CMOQR p8
A convex quadrilateral $ABCD$ is said to be dividable if for every internal point $P$, the area of $\triangle PAB$ plus the area of $\triangle PCD$ is equal to the area of $\triangle PBC$ plus the area of $\triangle PDA$. Characterize all quadrilaterals which are dividable.

2018 CMOQR p3
Let $ABC$ be a triangle with $AB = BC$. Prove that $\triangle ABC$ is an obtuse triangle if and only if the equation $$Ax^2 + Bx + C = 0$$has two distinct real roots, where $A$, $B$, $C$, are the angles in radians.

2018 CMOQR p4
Construct a convex polygon such that each of its sides has the same length as one of its diagonals and each diagonal has the same length as one of its sides, or prove that such a polygon does not exist.