geometry problems from Association Mathematical Quebec Competition (Canada) Secondary + College
with aops links in the names
collected inside aops here
2004-21 Secondary Competition
We consider a right triangle $ABC$, whose angle $B$ is right. Let $a, b$ and $c$ be the sidelengths opposite to the vertices $A, B$ and $C$ respectively. On the outside of the triangle, we construct the square $BHIC$ edge on side $a$, the square $ACDE$, on side $b$, and the square $AFGB$, on side $c$. Finally, we construct the line segments $ID, EF$ and $GH$. Prove that the areas of the triangles $AEF, BGH$ and $CID$ are equal to the area of triangle $ABC$.
In triangle $ABC$, we have $AB = 13, AC = 15$ and $BC = 14$.
Determine the point $E$ on $AB$ that makes $EFHG$ a square.
Consider six points randomly arranged on a circumference of radius $1$. Show that, whatever this arrangement, we are always able to choose three points, which we will call $A, B$ and $C$, so that the angle $ABC$ is less than or equal to $30$ degrees.
A ball bounces between two parallel walls while traversing the zigzag $ABCDEF$.
The distance $AA'$ between the two walls is $5$ meters. To his surprise, Professor Cosinus finds that, in his experience, the length of the zigzag is exactly equal to the sum of the distances $AA'$ and $A'F$. What is the length of the zigzag traversed by the ball?
Note: The size of the ball is not taken into account and it is considered as a geometric point. Note also that the figure has not been drawn to scale and therefore it is unnecessary to measure.
If we connect the vertices of a regular octagon together which have a neighboring vertex in common, we obtain in the center of the figure a new regular octagon, offset and smaller than the first (in gray on the drawing). If the area of the initial octagon is $1$, what is the area of the new octagon?
Divide a rectangle equal to $9$ cm long and $3$ cm wide into eight squares.
A flat egg-shaped figure is delimited by four arcs of circles, the arcs $AB, BF, FE$ and $EA$ as indicated in the figure below. Knowing that the radius $AO$ has length $1$, determine the area of the figure.
Three concurrent lines are joined to form triangles as in the figure. Calculate the sum of the angles $a,b,c,d,e$ and $f$.
Mr. A. Bajour makes lampshades in the shape of a truncated pyramid with a square base (see the first figure). The isosceles trapezoids forming the surface of its lampshades have the dimensions, in cm, given in the second figure. Each shade should be shipped in a box in the shape of a right prism with a square base that is as small as possible. What is the total area of such a box?
Note: the pyramid must be placed in the box so that its base touches the bottom of the box.
Four circles of radius $1$ are tangent to each other and to a large circle containing them, as in the figure . Calculate the area of the shaded area, inside the large circle but outside the small circles.
Two circles with radii of $9$ cm and $17$ cm are contained in a rectangle, one of the sides of which is $50$ cm. The two circles are tangent to each other and touch two adjacent sides of the rectangle, as in the figure. Calculate the area of the rectangle.
A line is located at $\sqrt2/2$ units from the center of a circle of radius $1$, separating it into two parts. What is the area of the smallest part ?
In a circle of radius $r$, two chords $AB$ and $CD $ intersect perpendicularly at $X$.
Show that $| XA |^2 + | XB |^2 + | XC |^2 + | XD |^2 = 4 r^2$.
Let $R$ be a rectangle of dimensions $a \times b$. What are the area and perimeter of all points located at a distance of $1$ or less from $R$ (including $R$)?
What is the radius of the circle circumscribed around a isosceles triangle with base $8$ and altitude $8$?
A dart is planted $12$ cm from the center of a circular target $20$ cm in radius. We launch a another dart, which hits the target (each point on the target is as likely as any other to be achieved). What is the probability that the second dart is closer to the first than to the edge of the target?
Suppose we have two concentric circles, as shown in the figure below. A chord of the outer circle has length $24$ m and is tangent to the inner circle. What is the area between the outer and inner circles?
In the right triangle $ABC$ with $\angle ABC=90^o$, $BH$ is altitude.
We know the following facts: $AH = (2x + 1)$ cm, $HC = (5x - 4) $cm, $BH = 3x$ cm.
Determine the numerical value of the perimeter of triangle $ABC$ (in cm)
What is the radius of the circle inscribed in the triangle of sides $16, 17, 17$ ?
Two lines intersect at a point $A_0$ and are separated by a angle of $6^o$. We take points $A_2, A_4, A_6,...$ on one of the straight lines and $A_1, A_3, A_5,...$ on the other, so that the segments $A_0 A_1, A_1 A_2, A_2 A_3, A_3 A_4,...$ all have the same length. What is the maximum number of distinct segments that can be place so?
A rectangular parallelepiped has sides of integer lengths chosen such that the sum of the number of its vertices, the length of its edges, the area of its faces and its volume is $2015$. What is the volume of the parallelepiped?
Let $ABCD$ be a square . Let be a point $P$ such that $PA = 1, P B = 2$ and $P C = 3$. Find the length of $PD$.
Consider a line segment and a point $C$ on this segment. There are two other points $A$ and $B$ on the segment on either side of $C$. Let there be the two circles whose diameters are $AC$ and $CB$ and whose the centers also lie be on the initial segment. $OF$ is tangent common to the two circles which meets them respectively in $D$ and $E$. Show that there is a unique circle passing points $A, D, E$ and $B$.
Let $I$ be the point of intersection of the three interior angle bisectors of a triangle. Prove that any line passing through$ I$ divides the perimeter of the triangle into two equal parts if and only if this line divides the area of the triangle into two equal parts.
Let be a triangle $\vartriangle ABC$ and parallel segments $PQ, RS$ and $TU$ to $AB, BC$ and $CA$, respectively. Suppose that these three segments, as shown in the figure below, meet at points $X, Y$ and $Z$. If each segment $PQ, RS$ and $TU$ subdivides the triangle $\vartriangle ABC$ into two parts of equal area, and if the area of the inner triangle $\vartriangle XY Z$ is $1$, then what is the area of $\vartriangle ABC$?
After New Years Eve, Justin's dad asked him to take the empty bottles of the guests to the convenience store. Year to facilitate transportation, Justin placed six of these bottles in a box he found, in which they just fit, with the configuration illustrated opposite (top view). Knowing that each bottle has a diameter of $10$ centimeters, which are the exact dimensions of the bottom of the box?
A triangulation is a cutting of a geometrical figure in a number of triangles. For example, the figure shows a triangulation of a square into four triangles. Prove that there is no triangulation of the square in just three triangles, such that these three triangles all have the same area.
In the following figure, we have shown the top of a hand spinner. Curves between two points are all arcs of a circle with a radius of $1$ cm. In their point of contact, the tangents of two curves coincide. In addition, the top is well balanced, that is, it has three axes of symmetry. Find the area represented by the figure.
Mathilde has two rectangular boxes of the same dimensions. These dimensions (height and width) are integer numbers of centimeters. The height of boxes is strictly greater than their width. Mathilde superimposes the two boxes as in the figure opposite. She then realizes that the area of the quadrilateral formed by the superposition of two boxes (in gray in the figure) is an integer number of centimeters . What is the minimum height that these boxes could have?
A flower containing $8$ petals is symmetrical on $8$ axes. In addition, its adjacent petals are tangent to each other. Each petal consists of two arcs of a circle radius $1$ cm. What is the area of the flower (the area of the gray area)?
As Douglas Adams points out in his novel "The Hitchhiker's Guide to the Galaxy ", the number $42$ is the answer to the ultimate question of life, the universe and of everything else. Having said that, calculate the area of $42$ knowing that each pup square dot is of dimension $1$ cm and that the curves are arcs of circles.
Fold a $12 \times 18$ sheet of paper in half, as shown below.
What is the area of the shape obtained after bending is performed, such as in the figure ?
Let be an equilateral triangle $ABC$ whose side is $3$ cm. We draw three arcs of a circle: the arc $BC$ of a circle with center $A$, the arc $CA$ of a circle with center $ B$, and the arc of a circle $AB$ of a circle with the center $C$. What is the total area of the resulting figure?
In the figure, the hatched segments are parallel to the $x$ axis. What is the area of the shaded quadrilateral?
2004-20 College Competition
Find a rectangle and an isosceles triangle with the same perimeter and same area and with all sides of integer lengths.
A rectangular glass $ABCD$ is supported on the corner of a balcony (see figure). The glass touches the horizontal ramps at points $P$ and $Q$ respectively and the vertical column at point $S$. The base $AB$ of the glass rests completely on the balcony floor. Knowing that $OP = OQ = 30$ cm, $OS = 15$ cm and $RO = 80$ cm, determine the dimensions of the glass.
An "eye" has the following form: 
where $AB, AC, A 'B', A 'C'$ are tangent to the circle of quadrilateral $B B' C'C$ with center $O$. Points $A, O$ and $A '$ are collinear. Knowing that $AB = AC = A 'B' = A 'C' = 1$ cm and that the area of the circle is $\pi / 4$ cm$^2$ , determine the distance between points $A$ and $A'$.
A baker wants to grow wheat on a $2$ km$^2$ field in the form of a "slice bread "(= rectangle surmounted by a semicircle, see figure). What dimensions will it need give to its field so that the fence delimiting it is as short as possible?
A triangle whose angles are $A, B$ and $C$ is inscribed in a circle of radius $R$.
What is the area of this triangle (in terms of $A,B,C,R$)?
Let $M$ and $N$ be circles with respective radii $m$ and $n$ (integers $\ge 1$). Suppose the circle $M$ rotates without sliding around circle $N$ (see figure) until point $A$ of $M$ returns to the point $B$ of $N$. How many turns around its center will circle $M$ make?
What is, on average, the terrestrial distance between two points chosen completely at random on Earth's surface ?
Notes: We suppose that the Earth is a perfect sphere of radius $R$ and it is a question of expressing the answer according to $R$.
The terrestrial distance between two points is the minimum length of an arc joining these two points traced on the earth's surface (arc of ''large circle'').
During a competition, a shooter received $50$ consecutive shots of a pistol at a square target of $70$ cm by $70$ cm. Prove that at least two of the hole centers on the target are within $15$ cm apart.
Consider an arbitrary triangle whose sides measure $a, b$ and $c$ units. The area of this triangle is given by the following formula due to the Greek mathematician Heron $S =\sqrt{p (p - a) (p - b) (p - c)}$,where $p=\frac{1}{2}(a+b+c)$ denotes the half-perimeter of the triangle. Show that Heron's formula entails the Pythagorean theorem , $a^2 + b^2 = c^2$ in the particular case where the triangle is given right with hypotenuse $c$.
A group of oceanographic researchers has three platforms, $A, B$, and $C$, on the Pacific Ocean, each containing a laboratory. Their research is limited to the portion of the ocean included in the spherical triangle $ABC$. Knowing that $\angle A = 90^o$, $\angle B = 45^o$, $\angle C = 60^o$, determine the total area of the spherical triangle $ABC$ (in terms of the area of the earth).
Note: Unlike the case of plane triangles, in a triangle spherical, the sum of the angles is not equal to $180$ degrees. She can vary by always being greater than $180$ degrees, as Illustrated in the following figure where the three angles are $90$ degrees each and their sum is $270$ degrees $> 180$ degrees.
Two circular pulleys are connected by a well-tensioned belt as in the following figure. The pulleys are $40$ cm and $10$ cm radii, their centers are $60$ cm apart. What is the length of the belt?'
Inside a right triangle, there is a quarter circle of radius $4$, a semicircle of radius $1$ and a small circle. The arcs of circles are tangent between them and tangent to the hypotenuse of the triangle as in the figure . Find the radius of the smallest circle.
Josée has a small cubic bottle of $3$ centimeters per side containing precious liquid that a merchant wants to buy from him. To fix the price, the merchant claims that the bottle is two-thirds full because if we place it so that a large diagonal of the cube (for example $AG$ in the figure) to be perpendicular to the ground, there is liquid up to two-thirds of that diagonal. In order to find out if Josée has an interest in accepting this interpretation, find the fraction of the volume of the vial that the liquid actually represents.
The Republic of Mathematics has decided to make a commemorative coin whose face side represents Gauss in an inscribed circle in a rectangle, itself inscribed in a more large circle of radius $R$. Express the perimeter of the rectangle as a function of $R$, if the area in gray at the outside of the small circle represents a fraction $f = \frac{4\sqrt3}{\pi} -1$ of the area of this small circle containing Gauss.
In a park, a square of side $\frac{9\sqrt{2}}{8}$ m is reserved for the arrangement of a rudimentary seesaw. This will be built using a $0.25$ plank m wide, as long as possible, the center of which will be placed on a block of $4-2\sqrt3$ m in diameter. After identifying the region of the square to use, the constructors installed the board and realized that when it hits the ground, a side or the other, it always passes through the same point $P$ whose distance from the ground is $\frac{\sqrt3}{3}$ m. In this context, calculate the length of the board knowing that the end of it must be entirely in the square at the time of its contact with the ground.
Emilie runs along a circular track with a radius of $100$ meters. Olivier has an urgent message to give her and arrives at the track just when she is diametrically opposed to him, but she does not notice him and continues on her path. Olivier runs at the same speed as Emilie and allows himself to go in a straight line in the direction he wants without following the circular track, show that the angle $\theta$ (measured, in radians, compared to the diameter that separates them initially) that he must choose to join as quickly as possible Émilie is such that: $\theta = \cos (\theta )$.
Pascale has a triangular object (scalene triangle $ABC$) made of translucent plastic with a uniform thickness of $5$ mm and containing a colored liquid. When she places the base AB $36$ cm horizontally, the liquid reaches a height of $3$ cm. When she places the base $BC$ of this object horizontally, the liquid reaches a height equal to $1/4$ of the height resulting from the vertex $A$ of the triangle. By placing the triangular object on the base $CA$, the height of the liquid is $4$ cm. What is the length of the base $CA$ ?
A company wants to create a shiny new logo to replace its old one outdoor sign. This will be formed by a golden rectangle of $1$ meter by $\frac{1+\sqrt5}{2}$ meter inscribed in a circle. In addition, each side of the rectangle will be the diameter of a semi-circle which will be added to the logo as shown in the figure. The company wants to stamp the rectangle and the lunulae (the shaded part) of its logo with a thin layer of $24$ karat gold. Knowing that $24$ karat thin foil gold cost 0.05 dollars per square centimeter, how much will it cost the company to restore its teaches ?
Let be a circle and a point $P$ which is outside it. From $P$ we draw lines which, when crossing the circle, define segments in the circle. Show that all midpoints of these chords are located on the same circle.
Jacqueline wants to calculate the area of triangle $NMI$ with right angle at $I$. She knows that the medians $MD$ and $NE$ intersect at $O$. She also knows that $OD=3$ and $OE=4$. What is the area of triangle $NMI$?
Show that for all regular polygons whose edge measures $2$ units, the area of the ring bounded by the circle circumscribed to the polygon and the circle inscribed in the polygon will always be equal to $\pi$.
Satellites used for GPS positioning can pinpoint any location on Earth. We program these so that they choose completely randomly $5$ places on our planet. If we assume that the Earth is a sphere, what is the probability that we can find a hemisphere (border included) on the surface of which at least $4$ of the chosen points?
To mark the Year of the Goat, Paula wants to build an enclosure for her goats hexagonal by alternating metal walls and wood walls. She already has the $3$ metal walls of lengths $1$ meter, $2$ meters and $9$ meters and she wants to use everything its wood, which makes it possible to obtain a total length of $18$ meters, to build the $3$ other walls. To make the corners in $A, B, C, D, E$ and $F$, she will use hinges with fixed opening which form an angle of $120^o$ and allow to join a wall of metal and a wood wall. Find out what the lengths of the timber walls will be his enclosure.
We can show that if two figures are similar, then the ratio of $\frac{perimeter^2}{area}$ is constant. But the converse is not true: it is possible that two figures have the same rato $\frac{perimeter^2}{area}$ are not similar. In this case, we say that the figures are almost similar. Find the dimensions of a rhombus that is almost similar like a rectangle of $2$ units of width and $3$ units of length.
An old circular porthole now has only two parallel bars, one of $22$ cm and the other of $18$ cm. As these are spaced $10$ cm apart and the new standards safety bars require that each opening have a maximum width of $9$ cm, we decide to add another parallel bar halfway between the two existing ones. What will be the length of this new bar?
Simon had a strange dream: the planetary council decided to separate North America and South America into two separate continents. The International Olympic Committee then took the opportunity to reshape the Olympic logo where the five colored rings were replaced by six colored discs. They opted for a figure where the six disks of the same size have their center on the same circle and where each disk is tangent with its two neighbors. When he woke up, Simon wanted to reproduce this logo in a giant format by painting each disc a different color. If the circle passing through the center of the six disks has a radius of $1$ meter, find what will be the total surface of the discs to be painted.
To move a massive rock of spherical shape of $4$ decimetres of radius, we use a cylindrical log of $4$ decimetres in diameter and a pole to produce leverage. By inserting the pole until it touches the ground, we realize that its other end comes to the same height as the rock (sectional view on the diagram below). The points of tangency of the rock and the log with the ground have distance $9$ decimetres . How long is the pole? We consider negligible the thickness of the pole.
Consider an ellipse with major axis $4$ cm and minor axis $3$ cm. Those axes are aligned on the diagonals of a square. The ellipse is also tangent to the square in four points. Find the area of the square.
Let be two small circles having radii of length $1$ and a large circle having a radius of length $\sqrt{10}-1$. The three circles are placed so that they are tangent externally in pairs two by two. Find the radius of the circle passing through the centers of the three circles.
Determine the coordinates of the center of the circle in the figure.
A dodecagon inscribed in a circle has six sides of length $a$ and six of length $b$, in any order. Let $C$ be a vertex adjacent to a side $AC$ of length $a$ and one side $CB$ of length $b$. Calculate the angle $\angle ACB$
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