Math Prize for Girls 2009-21 (MPfG) 57p

geometry problems from Math Prize for Girls Problems (1st round) and Olympiad (final round)

(MPfG)

with aops links in the names

The Olympiad Round started in 2010
Math Prize for Girls & Olympiad 2009-19 in pdf with solutions

it didn't take place in 2020

in 2021 only first round took place

                                            2010 - 2019 Olympiad (final round)

2011 Math Prize for Girls Olympiad p2

Let $\triangle ABC$ be an equilateral triangle. If $0 < r < 1$, let $D_r$ be the point on $\overline{AB}$ such that $AD_r = r \cdot AB$, let $E_r$ be the point on $\overline{BC}$ such that $BE_r = r \cdot BC$, and let $P_r$ be the point where $\overline{AE_r}$ and $\overline{CD_r}$ intersect. Prove that the set of points $P_r$ (over all $0 < r < 1$) lie on a circle.

2012 Math Prize for Girls Olympiad p1
Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that
$$\sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, .$$

2013 Math Prize for Girls Olympiad p2
Say that a (nondegenerate) triangle is funny if it satisfies the following condition: the altitude, median, and angle bisector drawn from one of the vertices divide the triangle into 4 non-overlapping triangles whose areas form (in some order) a 4-term arithmetic sequence. (One of these 4 triangles is allowed to be degenerate.) Find with proof all funny triangles.

2014 Math Prize for Girls Olympiad p1
Say that a convex quadrilateral is tasty if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.

2015 Math Prize for Girls Olympiad p2
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.

2016 Math Prize for Girls Olympiad p1
Triangle $T_1$ has sides of length $a_1$, $b_1$, and $c_1$; its area is $K_1$. Triangle $T_2$ has sides of length $a_2$, $b_2$, and $c_2$; its area is $K_2$. Triangle $T_3$ has sides of length $a_1 + a_2$, $b_1 + b_2$, and $c_1 + c_2$; its area is $K_3$.
(a) Prove that $K_1^2 + K_2^2 < K_3^2$.
(b) Prove that $\sqrt{K_1} + \sqrt{K_2} \le \sqrt{K_3} \,$.

2017 Math Prize for Girls Olympiad p3
Let $ABCD$ be a cyclic quadrilateral such that $\angle BAD \le \angle ADC$. Prove that $AC + CD \le AB + BD$.

2018 Math Prize for Girls Olympiad p1
Let $P$ be a point in the plane. Suppose that $P$ is inside (or on) each of 6 circles $\omega_1$, $\omega_2$, ..., $\omega_6$ in the plane. Prove that there exist distinct $i$ and $j$ so that the center of circle $\omega_i$ is inside (or on) circle $\omega_j$.

2019 Math Prize for Girls Olympiad p2

Let $ABC$ be an equilateral triangle with side length $1$. Say that a point $X$ on side $\overline{BC}$ is balanced if there exists a point $Y$ on side $\overline{AC}$ and a point $Z$ on side $\overline{AB}$ such that the triangle $XYZ$ is a right isosceles triangle with $XY = XZ$. Find with proof the length of the set of all balanced points on side $\overline{BC}$.

2009 - 2019, 2021 problems (1st round) 

2009 Math Prize for Girls problems p5

The figure below shows two parallel lines, $\ell$ and $m$, that are distance $12$ apart:

A circle is tangent to line $\ell$ at point $A$. Another circle is tangent to line $m$ at point $B$. The two circles are congruent and tangent to each other as shown. The distance between $A$ and $B$ is $13$. What is the radius of each circle?

2009 Math Prize for Girls problems p13

The figure below shows a right triangle $\triangle ABC$.

The legs $\overline{AB}$ and $\overline{BC}$ each have length $4$. An equilateral triangle $\triangle DEF$ is inscribed in $\triangle ABC$ as shown. Point $D$ is the midpoint of $\overline{BC}$. What is the area of $\triangle DEF$?

2010 Math Prize for Girls problems p6

The bases of a trapezoid have lengths 10 and 21, and the legs have lengths $\sqrt{34}$ and $3 \sqrt{5}$. What is the area of the trapezoid?

2010 Math Prize for Girls problems p11

In the figure below, each side of the rhombus has length 5 centimeters.

The circle lies entirely within the rhombus. The area of the circle is $n$ square centimeters, where $n$ is a positive integer. Compute the number of possible values of $n$.

2010 Math Prize for Girls problems p14

In the figure below, the three small circles are congruent and tangent to each other. The large circle is tangent to the three small circles. The area of the large circle is 1. What is the area of the shaded region?

2011 Math Prize for Girls problems p3
The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.
If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?

2011 Math Prize for Girls problems p5
Let $\triangle ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC = 5$. Let $I$ be the center of the circle inscribed in $\triangle ABC$. What is the product of $AI$, $BI$, and $CI$?

2011 Math Prize for Girls problems p6
Two circles each have radius 1. No point is inside both circles. The circles are contained in a square. What is the area of the smallest such square?

2011 Math Prize for Girls problems p8
In the figure below, points $A$, $B$, and $C$ are distance 6 from each other. Say that a point $X$ is reachable if there is a path (not necessarily straight) connecting $A$ and $X$ of length at most 8 that does not intersect the interior of $\overline{BC}$. (Both $X$ and the path must lie on the plane containing $A$, $B$, and $C$.) Let $R$ be the set of reachable points. What is the area of $R$?

2011 Math Prize for Girls problems p9
Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?

2012 Math Prize for Girls problems p2
In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2.
Adjacent circles are tangent to each other. What is the area of the shaded region?

2012 Math Prize for Girls problems p5
The figure below shows a semicircle inscribed in a right triangle.
The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?

2012 Math Prize for Girls problems p10
Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AC}$, and let $F$ be the midpoint of $\overline{AB}$. Let $G$ be the midpoint of $\overline{EC}$. One of the angles of $\triangle DFG$ is a right angle. What is the least possible value of $\frac{BC}{AG}$?

2013 Math Prize for Girls problems p1
The figure below shows two equilateral triangles each with area 1. The intersection of the two triangles is a regular hexagon. What is the area of the union of the two triangles?

2013 Math Prize for Girls problems p7
In the figure below, $\triangle ABC$ is an equilateral triangle. Point $A$ has coordinates $(1, 1)$, point $B$ is on the positive $y$-axis, and point $C$ is on the positive $x$-axis. What is the area of $\triangle ABC$?

2013 Math Prize for Girls problems p12
The rectangular parallelepiped (box) $P$ has some special properties. If one dimension of $P$ were doubled and another dimension were halved, then the surface area of $P$ would stay the same. If instead one dimension of $P$ were tripled and another dimension were divided by $3$, then the surface area of $P$ would still stay the same. If the middle (by length) dimension of $P$ is $1$, compute the least possible volume of $P$.

2013 Math Prize for Girls problems p15
Let $\triangle ABC$ be a triangle with $AB = 7$, $BC = 8$, and $AC = 9$. Point $D$ is on side $\overline{AC}$ such that $\angle CBD$ has measure $45^\circ$. What is the length of $\overline{BD}$?

2014 Math Prize for Girls problems p1
The four congruent circles below touch one another and each has radius 1.
What is the area of the shaded region?

2014 Math Prize for Girls problems p8
A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

2014 Math Prize for Girls problems p12

Let $B$ be a $1 \times 2 \times 4$ box (rectangular parallelepiped). Let $R$ be the set of points that are within distance 3 of some point in $B$. (Note that $R$ contains $B$.) What is the volume of $R$?

2014 Math Prize for Girls problems p14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?

2014 Math Prize for Girls problems p17
Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that $\frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x$ for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

2015 Math Prize for Girls problems p11 
Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.

2015 Math Prize for Girls problems p14
Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several nonoverlapping regions. Find the volume of the region that contains the center of $C$.

2015 Math Prize for Girls problems p20
In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$. A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$. Points $C$, $A$, and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE = 6$ and $FG = 9$. Find $AG$.

2016 Math Prize for Girls problems p1
Let $T$ be a triangle with side lengths 3, 4, and 5. If $P$ is a point in or on $T$, what is the greatest possible sum of the distances from $P$ to each of the three sides of $T$?

2016 Math Prize for Girls problems p8
A strip is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?

2016 Math Prize for Girls problems p15
Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.

2016 Math Prize for Girls problems p16
Let $A < B < C < D$ be positive integers such that every three of them form the side lengths of an obtuse triangle. Compute the least possible value of $D$.

2016 Math Prize for Girls problems p19

In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. The three lines $\ell_A$, $\ell_B$, and $\ell_C$ begin rotating clockwise about points $A$, $B$, and $C$, respectively. They rotate at the same angular rate. At any given time, the three lines form a triangle. Determine the largest possible area of such a triangle.

2017 Math Prize for Girls problems p2
In the figure below, $BDEF$ is a square inscribed in $\triangle ABC$. If $\frac{AB}{BC} = \frac{4}{5}$, what is the area of $BDEF$ divided by the area of $\triangle ABC$?

2017 Math Prize for Girls problems p12
Let $S$ be the set of all real values of $x$ with $0 < x < \pi/2$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$.

2017 Math Prize for Girls problems p17
Circle $\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$. Circle $\omega_2$ within $S$ with radius 2 is tangent externally to circle $\omega_1$ and is also tangent to line $a$. Circle $\omega_3$ within $S$ is tangent externally to both circles $\omega_1$ and $\omega_2$, and is also tangent to line $b$. Compute the radius of circle $\omega_3$.

2017 Math Prize for Girls problems p19
Up to similarity, there is a unique nondegenerate convex equilateral 13-gon whose internal angles have measures that are multiples of 20 degrees. Find it. Give your answer by listing the degree measures of its 13 external angles in clockwise or counterclockwise order. Start your list with the biggest external angle. You don't need to write the degree symbol $^\circ$

2018 Math Prize for Girls problems p4
Let $ABCDEF$ be a regular hexagon. Let $P$ be the intersection point of $\overline{AC}$ and $\overline{BD}$. Suppose that the area of triangle $EFP$ is 25. What is the area of the hexagon?

2018 Math Prize for Girls problems p10
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$.

2018 Math Prize for Girls problems p13
A circle overlaps an equilateral triangle of side length $100\sqrt{3}$. The three chords in the circle formed by the three sides of the triangle have lengths 6, 36, and 60, respectively. What is the area of the circle?

2018 Math Prize for Girls problems p15
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? (Recall that when light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.)

2018 Math Prize for Girls problems p17
Let $ABC$ be a triangle with $AB=5$, $BC=4$, and $CA=3$. On each side of $ABC$, externally erect a semicircle whose diameter is the corresponding side. Let $X$ be on the semicircular arc erected on side $\overline{BC}$ such that $\angle CBX$ has measure $15^\circ$. Let $Y$ be on the semicircular arc erected on side $\overline{CA}$ such that $\angle ACY$ has measure $15^\circ$. Similarly, let $Z$ be on the semicircular arc erected on side $\overline{AB}$ such that $\angle BAZ$ has measure $15^\circ$. What is the area of triangle $XYZ$?

2019 Math Prize for Girls problems p3
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.

2019 Math Prize for Girls problems p4
A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?

2019 Math Prize for Girls problems p13
Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.

2019 Math Prize for Girls problems p16
The figure shows a regular heptagon with sides of length 1.
Determine the indicated length $d$. Express your answer in simplified radical form.

2019 Math Prize for Girls problems p17
Let $P$ be a right prism whose two bases are equilateral triangles with side length 2. The height of $P$ is $2\sqrt{3}$. Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?

2021 Math Prize for Girls problems p3

Let $O$ be the center of an equilateral triangle $ABC$ of area $1/\pi$. As shown in the diagram below, a circle centered at $O$ meets the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect each of the triangle's sides. Compute the total area of all six shaded regions.

2021 Math Prize for Girls problems p8

In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$, what is the perimeter of $\triangle ABC$?

2021 Math Prize for Girls problems p12

Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?

2021 Math Prize for Girls problems p19

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points $m(P)$ where $P$ is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$?

source: https://mathprize.atfoundation.org/