geometry problems from School Mathematics Competition by University of New South Wales (Australia) with aops links
collected inside aops juniors + seniors
The competition is a three-hour open-book exam. Entrants are allowed to take any books and materials, but not computers with Internet connections, into the examination.
1964 - 2019, 2021
it didn't have geometry every year
Juniors
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $ {AB}$ (not $ ACB$) . Prove that $AD + BD = DC$.
$ABC$ is an equilateral triangle and $P$ a point inside the triangle. Perpendiculars are drawn from $P$ to the sides, the feet of the perpendiculars being $X, Y, Z$ respectively. Prove that $PX + PY + PZ$ is constant for all positions of $P$.
Show that this theorem can be extended to any convex polygon, (i. e. no interior angle greater than $180^o$), with equal sides. (Some perpendiculars may meet the sides produced).
Is the converse of the theorem also true, i. e. if the sum of the lengths of the perpendiculars to the sides of a convex polygon is constant, are the sides of the polygon necessarily equal?
If the converse theorem is true, prove it; if it is not, give an example to show why it is false.
Given any four points in the plane, no three of which lie on a straight line, prove that it is possible to choose three of them, $P_1, P_2$ and $P_3$ say, such that the circle through them either encloses, or goes through, the fourth point $P_4$.
How many different (non-congruent) triangles are there with sides of integer lengths and perimeter $24$? How many with perimeter $60$?
1968 UNSW Juniors p2 (same as 2007.3 Juniors)
In the triangle' $ABC$ let $AB > AC$ and let $D$ be the foot of the perpendicular from $A$ onto $BC$. Prove that: $AB - AC < BD -DC$.
(i) Which of the following four statements are true, and which are false? If a statement is false, give an example showing this. If the statement is true prove it.
(a) If a polygon inscribed in a circle has equal sides then it has equal angles.
(b) If a polygon inscribed in a circle has equal angles then It has equal sides.
(c) If a polygon has an inscribed circle (all its sides are tangent to the circle) and its sides are equal then its angles are also equal.
(d) If a polygon has an inscribed circle and its angles are equal then also its sides are equal.
(ii) Let $n$ be the number of sides of the polygon. For what values of $n$ are all four statements true?
1970 UNSW Juniors p5 (same as 2007.1 Juniors)
It is known that among all $n$-sided polygons inscribed in a circle, the regular n-gon has the greatest perimeter. Using this fact, prove that:
(i) if four points $A_1,A_2, A_3, A_4$ are chosen on a circle such that the sum of all six distances $A_iA_j$ is maximal then the points are vertices of a square.
(ii) if six points $A_1,A_2, A_3, A_4,A_5, A_6$ are chosen on a circle such that the sum of all fifteen distances $A_jA_j$ is maximal then the points are vertices of a regular hexagon.
(i) Prove that if $ABCD$ is a convex quadrilateral (i.e. every interior angle is less than $180^o$), the midpoints of its sides are the vertices of a parallelogram whose area is half of the area of $ABCD$.
(ii) $PQRS$ is a given parallelogram, $ABCD$ a convex quadrilateral such that $P$ is the midpoint of $AB, Q$ the midpoint of $BC, R$ the midpoint of $CD$, and S the midpoint of $DA$. Find the region in which $A$ must lie. Give reasons for your answer.
Let $D, E, F$ be the points of contact of the circle inscribed in the triangle $ABC$. Suppose that the triangle $DEF$ is similar to $ABC$. Prove that both $ABC$ and $DEF$ are equilateral.
(i) A rectangular billiard table, with pockets only at the four corners, has dimensions $5$ ft $\times 3$ ft. A ball is hit from a corner at an angle of $45^o$ to the sides. Prove that it will go into a pocket after several rebounds and find the number of rebounds.
It is of course assumed that the ball and pockets have conveniently small size and that the ball keeps rebounding at an angle of $45^o$ unless it hits a pocket.
(ii) Suppose that the billiard table is $6$ ft $\times 4$ ft and the ball starts $1$ ft away from a pocket on a $4$ ft side. Describe the path of the ball.
(iii) If in question (i), $5$ and $3$ are replaced by integers $p,q$ with no common factor, show that the ball goes into a pocket and find the number of rebounds.
(iv) Discuss the path of the ball if in question (ii), $6$ and $4$ are replaced by any positive integers $r$ and $s$.
Given a triangle with sides $a,b,c$ and a segment of length $d$. Denote by $p$ the smaller of $a$ and $d$, by $q$ the smaller of $b$ and $d$ and by $r$ the smaller of $c$ and $d$. Can you always construct a triangle with sides $p,q,r$?
Give reasons for your answer.
(i) Given a regular hexagon, $ABCDEF$, how many triangles can be formed such that each of their vertices is also a vertex of the hexagon?
(ii) Of the triangles in part (i) how many will have their centroid lying on the diagonal $AD$?
Give reasons for your answers.
It is possible to select three points in the plane so that each оf the points is at distance $1$ from the other two. (Take the vertices of an equilateral triangle with side $1$.) Find six points in the plane so that each point is at distance $1$ from exactly three of the other five points.
Given a triangle $ABC$ with acute angles, show how to find a point $P$ such that the four circumscribed circles around the four triangles $ABC, ABP, BCP, CAP$ have equal radii but do not coincide (that is, $P$ should not lie on the circumscribed circle of $ABC$). If you have constructed such a point $P$, prove that it has the required property.
Discuss the case when $ABC$ is not acute angled.
(i) $ABC$ is an isosceles triangle as shown. Show that the sum of the distances $PX$ and $PY$ is the same for all points $P$ on $BC$.
(ii) Conversely, suppose $ABC$ is a triangle, and that $P$ and $P'$ are distinct points on $BC$ such that $PX + PY = P'X' + P'Y'$. Show that $AB = AC$.
1979 UNSW Juniors p2 (same as 2006.5 Juniors)
Let $ABC$ be a right-angled triangle with hypotenuse $BC$ and with angle $ABC$ equal to $30^o$. Let $S$ be the centre of the inscribed circle, touching the three sides of the triangle as shown, and let $D$ be the midpoint of $BC$. Show that $AS = DS$.
Two spheres of radii $a,b$ ($a > b$) are glued together at a point solid so formed is rolled (without slipping) on a horizontal surface. The solid so formed is rolled (without slipping) on a horizontal surface. Show that $P$ describes a circle of radius $r$ where $r^2=\frac{16a^3b^3}{(a^2-b^2)}$.
In an arbitrary convex pentagon each side is translated outwards by $4$ units. Prove that the area is increased by at least $50$ square units.
Prove that there is no convex eight sided polygon with all angles equal and the sides distinct integers.
Let $P$ be a point inside the triangle $ABC$ and divide the triangle into six pieces as shown. The areas of four of the pieces are $40, 30, 35$ and $84$ as shown. Find the area of the triangle $ABC$.
Let $C_1$ and $C_2$ be circles with no points in common. Draw their common external tangents $L_1$ and $L_2$ and their common internal tangents, $\ell_1$ and $\ell_2$, as shown in the figure. Suppose that $L_1$ couches the circles at $A$ and $B$ and that $\ell_1$ intersects $L_1$ and $L_2$ at X and Y as shown. Prove that $AB$ and $XY$ are equal in length.
Prove that the angle at vertex $C$ of triangle $ABC$ is a right angle if and only if $\frac{1}{h^2}=\frac{1}{a^2}+\frac{1}{b^2}$ where $a, b$ are lentgths of the two sides from vertec $C$ and $h$ is the shortest distance from $C$ to the longest side $AB$.
A quadrilateral shaped frame has pivots at its corners and can freely move. Show that, if its diagonals are ever at right angles, then they are always at right angles.
Given four points $A,B,C,D$ in space, not all in a plane, show how to find a plane which is the same distance from all four points and has $A,C$ on one side and $B,D$ on the other side.
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?
$ABCD$ is a convex quadrilateral, and $M, N$ are the midpoints of the sides $AD$ and $BC$ respectively. If it is given that $MN = \frac12 (AB + CD)$ prove that the quadrilateral is a trapezium.
(a) A point is located outside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17$ units. Find the area of the square.
(b) As above, but the point is inside the square.
$ABCD$ is a parallelogram of unit area and $E, F, G, H$ are mid-points of the sides $BC, CD, DA, AB$ respectively. The line segments $AE,BF,CG$ and $DH$ dissect the interior of $ABCD$ into nine regions. Find the area of the central region.
The diagonals of a convex quadrilateral divide it into four triangles whose areas are $1,2,3 $ and $a$ units (in some order). Find all possible values of $a$.
Sixty seven points lie inside a regular hexagon with side length $2$ centimetres. Prove that a circular coin of radius $1$ centimetre can be placed to cover at least twelve of the points.
An old manuscript reads as follows: ... ''Having reached the island, walk from the palm tree to the white rock, turn $90^o$ right, and walk the same distance as you have just walked (that is, from the tree to the rock). Place a peg in the ground. Return to the palm tree, walk to the black rock, turn $90^o$ left, and walk a distance equal to that from the tree to the black rock. Place another peg in the ground. Dig for the treasure half way between the pegs.” When you arrive, you find that the rocks are easily identifiable, but many more trees have grown up and it is impossible to tell which one was meant. Can you find the treasure?
Are the following statements true or false? Prove your answers,
(a) A pentagon inscribed in a circle and having all of its angles equal must have all of its sides equal,
(b) A hexagon inscribed in a circle and having all of its angles equal must have all of its sides equal.
A square billiard table with side length $1$ metre has a pocket at each corner. A ball is struck from one corner and hits the opposite wall at a distance of $\frac{19}{94}$ metres from the adjacent corner. If the ball keeps travelling, how many walls will it hit before it falls into a pocket?
Four points are located in a plane. For each point, the sum of the distances to the other three is calculated; and these four sums are found to be the same. Determine all possible configurations of the four points.
$ABCD$ is a parallelogram, $X$ is a point on the diagonal $BD$. A line through $X$ parallel to $AB$ intersects $AD$ at the point $P$, a line through $X$ parallel to $BC$ intersects $AB$ at $Q$. Show that the area of the quadrilateral $APCQ$ is half the area of $ABCD$.
Let $\vartriangle ABC$ be right-angled. Let $A'$ be the mirror image of the vertex$ A$ in the side $BC$, let $B'$ be the mirror image of $B$ in $AC$ and $C'$ the mirror image of $C$ in $AB$. Find the ratio of areas $S_{ABC}:S_{A'B'C'}$.
A regular $21$-sided polygon is inscribed in a circle. Is it possible to choose five of its vertices in such a way as to define a pentagon, all of whose sides and diagonals have different lengths?
$ABCD$ is a trapezium in which $AB$ is parallel to $DC$, with $AB = BC = DA = 1$ and $CD = 1 + \sqrt2$. Let $E$ be a point on $AD$ such that we can fold the trapezium along a line passing through $E$ so that $A$ falls on $CD$. Find the maximum possible length of $DE$.
A long strip of paper of width $w$ is folded as shown in the figure. What is the smallest possible area of the overlapping triangle?
Suppose that a triangle $XYZ$ has side lengths $x, y, z$ with $s = (x + y + z)/2$. If the area of the triangle is denoted by $|XYZ|$, then Heron’s Formula states that $$|XY Z|^2 = s(s - x)(s - y)(s - z).$$ Let $OABC$ be a right-angled tetrahedron, with all angles at the vertex $O$ being right angles as
shown. Using Heron’s Formula or otherwise, prove the three dimensional Pythagoras’
Theorem:$|OAB|^2 + |OBC|^2 + |OCA|^2 = |ABC|^2$
Two circles, $C_1$ and $C_2$, with centers $O_1, O_2$, are externally tangent to each other at $T$. Their common tangents meet them at $A_1, A_2$ and $B_1, B_2$ respectively. Prove that the circles with diameters $A_1A_2$ and $B_1B_2$ are tangent to each other at $T$.
In the acute–angled triangle $ABC$, the perpendicular from $A$ onto $BC$ meets $BC$ at $D$, the perpendicular from $C$ onto $AB$ meets $AB$ at $E$. The length of $AE$ is $5$, the length of $BE$ is $3$, the length of $BD$ is $2$ and the length of $CD$ is $x$. Find $x$.
A cow is inside a square field. Its distances from the nearest three corners are $30$ m, $40$ m and $50$ m. What is its distance to the furthest corner, and what is the size of the field?
The convex quadrilateral $ABCD$ is divided into four smaller quadrilaterals by two straight lines joining the midpoints of opposite sides. Denote the areas of the four small quadrilaterals by $Q_1, Q_2, Q_3, Q_4$ in the order shown. Prove that $Q_1 + Q_3 = Q_2 + Q_4$.
A triangle is either isosceles (which for the purpose of this question we shall take to include equilateral) or scalene (all sides different).
(a) How many triangles are there with integer-length sides and perimeter $24$ ?
(b) How many triangles are there with integer-length sides and perimeter $36$ ?
(c) How many isosceles triangles are there with integer-length sides and perimeter $12n$ (where $n$ is an integer) ?
(d) How many triangles are there altogether with integer-length sides and perimeter $12n$ (where $n$ is an integer) ?
A disc of radius $1$ unit is cut into quadrants (identical quarters), and the quadrants are placed in a square of side $1$ unit.
(i) What is the least possible area of overlap?
(ii) What are the other possible areas of overlap?
A cubic box, side $1.2$ m, is placed on the ground next to a high wall. A ladder, length $3.5$ m, leans against the wall, and just touches the top edge of the box. How far from the box is the foot of the ladder, and how far above the box does the ladder touch the wall?
An American football field is $100$ yards long, and its width is half the average of its length and its diagonal. Find its area.
2006 UNSW Juniors p5 (same also as 1979 Juniors p2)
Two circular discs, of radii $r$ and $r\sqrt3$ respectively, are placed on a plane in such a way that their edges cross at right angles. Find the area of their overlap (that is, intersection), and the area covered by their union.
2007 UNSW Juniors p1 (same also as 1970 Juniors p5)
You are given nine square tiles, with sides of lengths $1, 4, 7, 8, 9, 10, 14,15$ and $18$ units, respectively. They can be used to tile a rectangle without gaps or overlaps.
Find the lengths of the sides of the rectangle, and show how to arrange the tiles.
2007 UNSW Juniors p3 (same also as 1968 Juniors p2)
A rectangular room is paved with square tiles all the same size.
Show how you can draw a right-angled triangle on the floor with the following properties:
The vertices of the triangle are at corners of tiles, the hypotenuse lies along the edge of the room, and the ratio of the lengths of the shorter sides is $2 : 3$.
Can it be done if the ratio of the lengths of the shorter sides is $m : n$?
An astronaut plants a flagpole on the surface of the Moon until the top of the flagpole is at eye-level height, at $1.5$ metres, and then the astronaut walks off over level ground until the top of the flagpole is just visible on the horizon. The astronaut uses a laser distance meter to measure the straightline distance from his line of sight to the top of the flagpole at $4.6$ kilometres. What is the radius of the Moon?
Let $ABC$ be a right triangle with right angled at $A$. Let $D, E$ be points on $BC$ with $BD = DE = EC$. Prove that $AD^2 + AE^2 = \frac59 BC^2$.
A bowl in the shape of a conical frustum is placed out in the rain at the start of a downpour. At the end of the downpour the water level $r$ is equal to (i) half the height of the bowl, (ii) the radius of the base of the bowl, and (iii) half the radius of the top of the bowl.
The volume of a cone is one-third the area of the circular base times the height. What was the reported rainfall over the catchment area?
In the figure below $AC = c$ and the points $E$ and $F$ lie on the line$ DG$ with $DE = a$ and $F G = b$. Show that if the area of the triangle $ABC$ is equal to the area of the rectangle $ADGC$ then $a + b =\frac{c}{2}$.
A person of height $1.7$ metres leaves a tall building at ground level and walks in a straight line direction up a path of constant gradient. He walks under a tall billboard after twenty metres and continues walking up the path for another five metres at which point he turns around and notices that the top of the billboard aligns horizontally with the top of the building. Then continues along up the path a further ten metres where he turns around again and notices that the top half of the building is now visible above the billboard. The height of the building is much greater than the height of the billboard which is much greater than the height of the person. What is the height of the building?
a) Prove that the radius of the inscribed circle to the triangle $\vartriangle ABC$ is given by $r =\frac{2S}{AB + BC + AC}$, where $S$ is the total area of the triangle $\vartriangle ABC$.
b) In a right-angled triangle, we draw the altitude onto the hypotenuse. This process is repeated in the two smaller right-angled triangles so formed and the process is then continued $2014$ times, as shown in the diagram. A circle is inscribed in each of the resulting $2^{2014}$ triangles. Find the total area of these circles.
Show how to cut a square of side length $1$ by straight lines, so that the resulting pieces can be assembled to form a rectangle in which the ratio of sides is $3 : 1$.
A triangle $\vartriangle ABC$ has squares $ABMP$ and $BCDK$ built on its outer sides. Prove that the median $BE$ of the triangle $\vartriangle ABC$ is also an altitude of the triangle $\vartriangle BMK$.
A point $A$ lies outside of a circle. Let $B$ be any point on the circle and $M$ be the midpoint of $AB$. As $B$ varies, describe in detail the curve traced by the point $M$.
Two straight lines pass through two vertices of a triangle such that the triangle is cut into four smaller pieces: three triangles and a quadrilateral. It is possible to choose the lines such that the areas of these pieces are the same?
There is a wolf at the centre of a square block of land. There is a dog located at each of the four vertices of the square. The wolf is allowed to move freely within the square and the dogs are only allowed to run by the sides of the square. Every dog is $50\%$ faster than the wolf. A dog alone is unable to stop the wolf. On the other hand, the wolf cannot pass if met by any two dogs. Find the strategy for the dogs to ensure that the wolf does not escape the square.
Two points $A$ and $B$ are chosen in the plane. Find the set of all points $M$ such that the $AM : BM = 2 : 1$.
At a theme park there is a game where contestants throw a circular disk with diameter $3.8$ cm onto a rectangular floor completely tiled with $4$ cm by $4$ cm squares. A prize is awarded if the disc lands completely inside a tile. Assuming that the division lines between tiles are infinitely thin, what is the probability of success?[/quote]
A bug sits at a corner of a rectangular box. The dimensions of the box are $1 \times 1 \times 2$. The bug can only travel on the faces and edges of the box. Find the shortest path for the bug to travel from its corner to the diagonally opposed corner.
Three grasshoppers play the following leapfrog game. They start off at three vertices of a square. At each step, a grasshopper leaps over another one and lands at the point symmetric to the point where it was. That is, if a grasshopper at a point $P$ leaps over a grasshopper at a point $Q$, then it lands at a point $R$ where $Q$ is the midpoint of $P R$. Is it possible for one of them to reach the fourth vertex of the square?
Seniors
A mountain is of a perfectly conical shape. The base is a circle of radius $2$ miles, and the steepest slopes leading up to the top are $3$ miles long. From a point $A$ at the southernmost point of the base a path leads up on the side of the mountain to $B$, a point on the northern slope and $2/5$ of the way to the top, (i. e. if $T$ is the top and $CT$ the northernmost slope, $C$ being on the ground, then $BC = 2/5 \,CT$).
If $AB$ is the shortest path leading along the mountainside from $A$ to $B$, find
(i) the length of the whole path $AB$,
(ii) The length of the part of the path between $P$ and $B$ where $P$ is a point on the path where it is horizontal.
$Q_1$ is a convex quadrilateral (no re-entrant angle) and $P$ is an interior point. $Q_2$ is the pedal quadrilateral of $Q_1$ with respect to $P$; i.e. the vertices of are the feet of the perpendiculars dropped from $P$ to the sides of $Q_1$ . In this way can be constructed a sequence of convex quadrilaterals, $Q_1, Q_2, Q_3, Q_4$ and $Q_5$ each being the pedal quadrilateral of its predecessor with respect to $P$. Prove that $Q_5$ is similar to $Q_1$.
If $r$ is the circumradius of a triangle $ABC$ then for each point $P$ within the triangle the smallest of the distances $AP, BP, CP$ is $\le r$.
What can you say about the truth of the statement: "the largest of the distances $AP, BP, CP$ is $\ge r$"?
Formulate and prove similar statements concerning the radius of the inscribed circle.
$P$ is a given point lying within the arms of an acute angle $AOB$ . Show how to construct the straight line through $P$ which cuts off the triangle of minimum area.
(i) Prove that no $3$ diagonals of a regular heptagon ($7$-sided polygon) are concurrent at a point other than a vertex of the heptagon. A diagonal is a line connecting two non-adjacent vertices.
(ii) How many points of intersection of pairs of diagonals lie within the heptagon?
(iii) Into how many compartments is the heptagon dissected by the diagonals?
(iv) Assuming that for $n$ odd no $3$ diagonals of a regular $n$-gon are concurrent, generalize (ii) and (iii) for the regular $n$-gon.
Suppose you are provided with a straight edge with two marks on it, one unit apart. The only operations allowable with this instrument are (a) ruling straight lines and (b) marking a point on a given straight line at a unit distance from a given point on the line. (You are not given a compass.) Using this instrument show how to perform the following constructions:
(i) Bisect a given angle,
(ii) Construct a square whose diagonals meet at a given point,
(iii) Construct a square with one vertex at a given point.
(i) Prove that any convex polygon with $n$ sides (where $n \ge 3$) can be dissected into $3(n - 2)$ cyclic quadrilaterals. (The opposite angles of a cyclic quadrilateral add up to $180^o$.)
(ii) Prove that every convex quadrilateral $ABCD$ can be dissected into four cyclic quadrilaterals by showing that:
(a) if $\angle BAD \ge \angle BCD$, there exists a point $X$ on the diagonal $AC$ such that $\angle BXD = 180^o - ( \angle BAD - \angle BCD)$.
(b) if $R$ is the point on $AD$ such that $\angle DXR = \angle DCX$, there are points $S, T, U$ on $AB, BC, CD$ respectively such that $RXSA, SXTB, TXUC$ and $UXRD$ are all cyclic quadrilaterals. (A polygon is called convex if all its interior angles are less than $180^o$.)
(iii) Discuss which quadrilaterals can be dissected into two cyclic quadrilaterals.
The triangle $ABC$ is isosceles and has a right angle at $B$. The point $M$ is on the circumcircle of $ABC$. Find the position of all points $M$ such that is possible to construct a triangle from the segments $BC, MA, MC$.
(a) Prove that in a triangle the lengths of altitude, angular bisector and median drawn from the same vertex follow each other in the same order.
(b) Assume further the triangle to be acute angled. Draw the circumscribed circle. Extend each of the above lines to meet the circle. Prove that the new lengths are still in the same order.
(c) Consider the extended length of the angle bisector. Prove that it is larger than the arithmetic mean of the two sides comprising the angle which has been bisected.
If the midpoint of each of the sides of a parallelogram is joined to the two opposite vertices, the eight lines so formed determine an octagon. Prove that the area of the octagon is $1/6$of the area of the parallelogram.
1978 UNSW Seniors p5 (same as 2007.5 Seniors)
Given four points in the plane, not on the same circle and no three on a line, how many circles $K$ are there with the property that $K$ is equidistant from all four points?
(The distance between a circle and a point $P$ is defined as follows: Draw the ray from the centre of the circle through the point $P$. This ray intersects the circle in a point $Q$. The distance $PQ$ is the distance between the circle and the point $P$.)
Suppose that the midpoints $P, Q$ and $R$, of the three altitudes of a triangle fall on a line. Show that one of the angles of the triangle must be a right-angle.
This is is a $2$ dimensional problem, i.e. it is a problem in the plane. The effective length $b$ of a pipe is the distance between the end points. Find the rigid pipe of longest effective length that can be taken around the corner in the above corridor. Is there a unique shape such a piece of pipe must have?
The circumference of a circle is divided into four arcs. Show that two of the line segments joining the mid-points of these arcs are at right angles to each other.
The surface of a cylinder consists of one curved and $2$ flat sections whereas that of a cone consists of one curved and one flat section. Suppose a right cone and a cylinder have a common circular face and the vertex of the cone is the centre of the opposite face of the cylinder. Suppose the ratio of their surface areas is $7:4$. Find the ratio of the length of the cylinder to its base radius.
Let $ABCD$ be a convex quadrilateral, and draw equilateral triangles $ABM, CDP, BCN, ADQ$ to the sides, the first two outwards, the other two inwards. Prove that $MN = AC$. What can you say about the quadrilateral $MNPQ$?
(i) A point $P$ is chosen at random on a circular disc. What is the chance that the chord with $P$ as mid-point is longer than the radius?
(ii) Diameters $AB, CD$ of a circle are chosen at random. What is the chance that both $AC, AD$ are longer than the radius?
A square is bisected by each of nine lines into two quadrilaterals so that the area of one of these quadrilaterals is twice the area of the other. Prove that there is a point which lies on at least three of these lines.
$ABC$ is a triangle, right angled at $C$. Let $CD$ be perpendicular to $AB$. The bisector of $ \angle CDB$ meets $CB$ in $X$ and the bisector of $\angle ADC$ meets $AC$ in $Y$. Prove that $CX = CY$.
The lengths of the sides of a triangle are in arithmetic progression and the greatest angle exceeds the least angle by $90^o$. Find the ratio of the lengths of the sides.
Two circles are tangent at a point $P$ and $A$ is a point on one of the circles. The tangent to this circle at $A$ intersects the other circle at the points $Q$ and $R$. Show that $A$ is equidistant from the lines $PQ$ and $PR$.
Equilateral triangles $A'BC, B'CA, C'AB$ are drawn external to the triangle $ABC$. Show that $AA', BB', CC$' are equal and that they intersect in a common point.
Let $ABC$ be a triangle and $X,Y, Z$ points on the sides $BC, CA, AB$ respectively. Show that if $BX \le XC$, $CY \le YA$, $AZ \le ZB$, then the area of triangle $XYZ$ is not less than one quarter of the area of triangle $ABC$. Show also that, in any case, one of the corner triangles $AZY, BXZ, CYZ$ has area not greater than the area of triangle $XYZ$.
1989 UNSW Seniors p2 (same as 1997.1 Seniors)
Given $3$ points $M, N, P$ not in a straight line, show how to construct a triangle $ABC$ such that $M$ is the midpoint of side $AB, N$ is a point on $BC$ such that $BN = \frac14 BC$ and $P$ is on $AC$ such that $CP= \frac14 CA$.
i) Given three non-collinear points $E, F, G$ construct a fourth point $H$ such that there can be found a quadrilateral $ABCD$ having $E,F,G,H$ as the midpoints of $AB,BC,CD,DA$ respectively.
ii) Given a convex pentagon $UVWXY$ show how to construct five points $P, Q, R, S, T$ having $U,V,W,X,Y$ as midpoints of $PQ,QR,RS,ST$ and $TP$ respectively.
Through a point inside a triangle of area $A$ are drawn three lines parallel to the sides of the triangle. These lines partition the interior of the triangle into three parallelograms and three triangles. If the triangles have areas $A_1,A_2$ and $A_3$, prove that $\sqrt{A_1}+ \sqrt{A_2}+\sqrt{A_3}= \sqrt{A}$
The sum of the lengths of the twelve edges of a rectangular box is $24$ metres, and the sum of the areas of the six faces is $18$ square metres. What is the largest possible volume of the box?
A small circle is located inside a larger circle, with the two circles touching at the point $A$. $P$ is a point on the large circle, and $T$ is a point on the small circle such that $PT$ is tangent to the small circle. Prove that provided $P \ne A$, the ratio of the lengths of $PT$ and $PA$ is the same for any point $P$ on the large circle.
Show that is possible for a cube and a plane to intersect in a regular hexagon, but impossible for a cube and a plane to intersect in a regular pentagon.
In a circle, $AB$ and $CD$ are two chords, perpendicular to each other and intersecting at $P$. The perpendicular from $P$ to $BC$ meets $BC$ at $X$. When $XP$ is extended it meets $AD$ at $Y$ . Show that $Y$ is the midpoint of $AD$.
Convex hexagon $ABCDEF$ is inscribed in a circle. If its diagonals $AD, BE$ and $CF$ meet at one point, prove that $AB \cdot CD \cdot EF = BC\cdot DE \cdot FA$.
Let $A$ be a point outside a circle $C$. For any point $P$ on $C$, let $Q$ be the vertex opposite $A$ on the square $APQR$. Determine the path traced out by $Q$ as $P$ moves around the circle $C$.
In a triangle $ABC$, $\angle B= 2\angle A$. Let $a = | BC |$, $b = | CA |$ and $c = | AB |$. Prove that $ac = b^2 - a^2$.
A circle on diameter $AB$ is given, together with a point $P$ inside the circle but not on $AB$. Show how to construct, using only an unmarked ruler, a line through $P$ perpendicular to $AB$. Prove that your construction succeeds.
A pentagon is said to be regular if its five sides all have equal length and if all its angles are equal.
(i) Show that a cube can be cut by a plane in such a way that the cross–section is a pentagon.
(ii) Prove that the pentagon is not regular.
A convex $12$-sided polygon is inscribed in a circle. Six of its sides have length $\sqrt2$ and six have length $\sqrt{24}$. What is the radius of the circle?
Two circles, of centres $O_1$ and $O_2$, intersect at $A$ and $B$. Through $A$ and $B$ are drawn two parallel lines which cut the two circles forming a quadrilateral $WXYZ$.
(a) Show that $WXYZ$ is a parallelogram, and that the length of one of its sides is equal to the length of the chord $AB$.
(b) Show that the area of $WXYZ $ is maximal when $WX$ is parallel to $O_1O_2$.
$A, B, C, D$ are points on a straight line (in the given order) such that $BC = 2AB, CD = AC$. Draw a circle through $A$ and $C$ and another circle through $B$ and $D$. Prove that the common chord of the two circles intersects the given line in the midpoint of $AC$.
A polytope $P$ in the plane is called a lattice polytope if all its vertices lie on integral lattice points:
those of the form $(n, m)$ with $n, m$ integers. It is known that there exist rational numbers $a, b$
and $c$ such that for any lattice polytope $P$ its area is area $(P) = aE + bI + c$
where $E$ is the number of lattice points lying on the edges of $P$ and $I$ is the number of lattice
points lying inside $P$. [Thus in the example $E = 6, I = 4$].
(i) Find $a, b$ and $c$.
(ii) Prove the formula is valid for any lattice triangle.
$UNSW $is a square. The points $A$ and $B$ lie on $UN$ and $UW$ respectively and $UA = UB$. Also, $E$ lies on $BN$ and $UE$ is perpendicular to $BN$. Prove that $\angle SEA = 90^o$.
Let $OX, OY$ denote the positive $x$ and $y$ axes. The rigid triangle $ABC$ has a rightangle at $C$. It glides in the plane, $A$ on $OY$ , $B$ on $OX$, while $C$ is on the opposite side of $AB$ to $O$.
(i) What is the path described by the midpoint of $AB$?
(ii) What is the path described by $C$?
$AB$ and $AW$ are tangents to a circle from an external point $A$. $UNA$ is a secant through $A$, cutting the circle at $U$ and $N$. $WS$ is a chord of the circle with $UN\parallel SW$. $SB$ intersects $UN$ at $C$. Prove that $UC = CN$.
A triangle with sides $13, 14$ and $15$ sits around the top half of a sphere of radius $5$ (that is, it touches the sphere at three points, as shown in the diagram above). How far is the plane of the triangle from the centre of the sphere?
In an alleyway with tall buildings on both sides, a ladder of length $3.9$ m leans from the foot of the west wall on to the east wall, while a ladder of length $2.5$ m leans the other way across the alleyway, from the foot of the east wall on to the west wall. Looking north along the alleyway, the ladders appear to cross $1\frac27$ m above the roadway. How wide is the alleyway?
A spider is sitting on the end wall of a room that is $4$ m wide, $6$ m long and $5$ m high. He is $1$ m from the ceiling, and $2$ m from each of the side walls. He spies his dinner, a clever but lazy fly, sitting at the other end of the room, $1$ m from the floor and $2$ m from each side wall.
The spider could simply run up to the ceiling, along the ceiling and down the opposite wall to catch his dinner, but he is a clever fellow, and realises that he can reach his target by a shorter route! Can you find the length of his shortest route to the fly?
The fly, on the other hand, is not only clever enough to see how the spider can get to her by the shortest possible route, but also realises that if she walks a little way towards the floor, she can get further away from the spider. How far should she walk in order to get as far as possible away from the spider, and now how far is she from the spider by his shortest possible route?
Four ants are situated at the corners of a square. Each one faces the next one anticlockwise around the square. They all start moving at the same time, and continue walking directly towards the next ant. Clearly, their paths spiral in to the centre of the square. How far does each ant move along its path?
(a) Given an isosceles trapezium, with equal sides of length $a$, parallel sides of lengths $b$ and $c$, and diagonal of length $d$, prove that $d^2 = a^2 + bc$.
(b) Hence, or otherwise, find the (shortest) distance across the surface of the Earth from London ($52^o$ N, $0^o$ E) to Sydney ($35^o$ S, $152^o$ E), assuming the Earth is a sphere of circumference $40000$ Km.
A triangle has one side that is the space diagonal (or long diagonal) of a cube and the other two sides lie on the surface of the cube whose side has length $a$.
1. What is the smallest perimeter that such a triangle can have?
2. What is the smallest area that such a triangle can have?
Let $ABC$ be a triangle with acute angles at $B$ and $C$. For any $X$ on $BC$, let $M$ and $N$ be the feet of the perpendiculars from $X$ to $AB$ and $AC$. Show how to find $X$ so that $MN$ is parallel to $BC$.
Consider the triangle shown below with vertices $A, B, C$ where point $D$ lies on the side $AB$, point $E$ lies on the side $BC$ and point $F$ lies on the side $AC$, and the three lines $AE, BF$ and $CD$ intersect at a common point $G$. Show that$$\frac{Area(\vartriangle CGF)}{Area(\vartriangle AGF)} =\frac{Area(\vartriangle BGC)}{Area(\vartriangle BGA}$$
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 $\frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}$.
In this problem you may assume the following result: Ptolemy’s Theorem:
If $ABCD$ is a cyclic quadrilateral, that is a quadrilateral inscribed in a circle, then $BD \cdot AC = AD \cdot BC + AB \cdot DC$.
Suppose ABCDE is a regular pentagon inscribed in a circle. Let P be any point on the arc BC. Prove that $PA + PD = PB + PC + PE$
Given two circles of radius $1$ with their centres one unit apart, a point $A$ is chosen on the first circle. Two other points $B_1$ and $B_2$ are chosen on the second circle, so that they are symmetric with respect to the line connecting the centres of the circles. Prove that $(AB_1)^2 + (AB_2)^2 \ge 2$.
The radius of the circumscribed circle of a triangle is $65/6$ . Find the third side of the triangle if the other two are $20$ and $13$ and every angle is acute.
Let $A$ be a point on the circle $C_1$ and $B$ be a point on the circle $C_2$ as shown. Let $M$ be the midpoint of $AB$. As $A$ and $B$ move independently around each circle, describe in detail the curve traced by the point $M$.
There is a rabbit in the centre of a square block of land. There is a wolf located at each of the four vertices of the square. The rabbit is allowed to run freely within the square and the wolves are only allowed to run by the sides. Every wolf is $40\%$ faster than the rabbit. Is there a strategy for the rabbit to escape the square?
Let $ABCD$ be a parallelogram and let the bisector of the angle $\angle BAD$ intersect the side $BC$ and the side $CD$ in the points $K$ and $L$, respectively. Prove that the centre of the circle through the points $C, K$ and $L$ lies on the circle through the points $B, C$ and $D$.
2018 UNSW Seniors p6 (Simson Line)
Let 4ABC be a triangle and S be the corresponding circumscribed circle. Let Q be a
point on S. Prove that the bases of perpendiculars dropped from Q to the sides of the
triangle 4ABC lie on a straight line.
Let $\vartriangle ABC$ be a triangle and let every angle of $\vartriangle ABC$ be acute. Prove that$$R \le \frac{ AB + BC + CA}{4}$$where $R$ is the radius of the circumscribed circle.
An ant crawls on a table with constant speed in one direction, then every $15$ minutes changes direction by turning $90^o$. Prove that the ant can only return to the original position after a whole number of hours have elapsed.
An island inhabitants design an emergency response service based on helicopters. Their island is a disc of radius $100$ km and they plan to purchase helicopters capable of flying $300$ km/h. Find the minimum number helicopters needed for the emergency service in order to be able to reach every point of the island within $10$ min. Assume that helicopter’s takeoff and landing times are negligible.
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