geometry problems from USA Mathematical Talent Search (USAMTS)
with aops links in the names
Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$.
(a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$.
(b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.
2011 USAMTS Round 2 p4
A luns with vertices $X$ and $Y$ is a region bounded by two circular arcs meeting at the endpoints $X$ and $Y$. Let $A$, $B$, and $V$ be points such that $\angle AVB=75^\circ$, $AV=\sqrt{2}$ and $BV=\sqrt{3}$. Let $\mathcal{L}$ be the largest area luns with vertices $A$ and $B$ that does not intersect the lines $VA$ or $VB$ in any points other than $A$ and $B$. Define $k$ as the area of $\mathcal{L}$. Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]
2012 USAMTS Round 1 p2
Three wooden equilateral triangles of side length $18$ inches are placed on axles as shown in the diagram to the right. Each axle is $30$ inches from the other two axles. A $144$-inch leather band is wrapped around the wooden triangles, and a dot at the top corner is painted as shown. The three triangles are then rotated at the same speed and the band rotates without slipping or stretching. Compute the length of the path that the dot travels before it returns to its initial position at the top corner.
2012 USAMTS Round 2 p5
A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.
2012 USAMTS Round 3 p3
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
2013 USAMTS Round 1 p5
Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?
2013 USAMTS Round 2 p2
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
2013 USAMTS Round 3 p3
Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point.
2014 USAMTS Round 1 p4
Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.
2014 USAMTS Round 2 p3
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.
2014 USAMTS Round 3 p2
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
2015 USAMTS Round 3 p4
Let $\triangle ABC$ be a triangle with $AB<AC$. Let the angle bisector of $\angle BAC$ meet $BC$ at $D$, and let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $B$ to $\overline{AD}$. Extend $\overline{BP}$ to meet $\overline{AM}$ at $Q$. Show that $\overline{DQ}$ is parallel to $\overline{AB}$.
2016 USAMTS Round 1 p5
Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.
2016 USAMTS Round 2 p3
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular $n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum possible value of $d$?
2016 USAMTS Round 3 p4
Let ${A_1, \dots , A_n }$ and ${B_1, \dots , B_n}$ be sets of points in the plane. Suppose that for all points $x$, $D \left( x , A_1 \right) + D \left( x , A_2 \right) + \cdots + D \left( x , A_n \right) \ge D \left( x , B_1 \right) + D \left( x , B_2 \right) + \cdots + D \left( x , B_n \right)$ where $D \left( x , y \right)$ denotes the distance between $x$ and $y$. Show that the $A_i$'s and the $B_i$'s share the same center of mass.
1998 USAMTS Round 1 p
1998 USAMTS Round 2 p
1998 USAMTS Round 3 p
1999 USAMTS Round 1 p
1999 USAMTS Round 2 p
1999 USAMTS Round 3 p
2000 USAMTS Round 1 p
2000 USAMTS Round 2 p
2000 USAMTS Round 3 p
2001 USAMTS Round 1 p
2001 USAMTS Round 2 p
2001 USAMTS Round 3 p
2002 USAMTS Round 1 p
2002 USAMTS Round 2 p
2002 USAMTS Round 3 p
2003 USAMTS Round 1 p
2003 USAMTS Round 2 p
2003 USAMTS Round 3 p
2004 USAMTS Round 1 p
2004 USAMTS Round 2 p
2004 USAMTS Round 3 p
2005 USAMTS Round 1 p
2005 USAMTS Round 2 p
2005 USAMTS Round 3 p
2006 USAMTS Round 1 p
2006 USAMTS Round 2 p
2006 USAMTS Round 3 p
2007 USAMTS Round 1 p
2007 USAMTS Round 2 p
2007 USAMTS Round 3 p
2008 USAMTS Round 1 p
2008 USAMTS Round 2 p
2008 USAMTS Round 3 p
2009 USAMTS Round 1 p
2009 USAMTS Round 2 p
2009 USAMTS Round 3 p
2010 USAMTS Round 1 p
2010 USAMTS Round 2 p
2010 USAMTS Round 3 p
with aops links in the names
2011-2016
the rest years under construction
2011 USAMTS Round 1 p4the rest years under construction
Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$.
(a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$.
(b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.
2011 USAMTS Round 2 p4
A luns with vertices $X$ and $Y$ is a region bounded by two circular arcs meeting at the endpoints $X$ and $Y$. Let $A$, $B$, and $V$ be points such that $\angle AVB=75^\circ$, $AV=\sqrt{2}$ and $BV=\sqrt{3}$. Let $\mathcal{L}$ be the largest area luns with vertices $A$ and $B$ that does not intersect the lines $VA$ or $VB$ in any points other than $A$ and $B$. Define $k$ as the area of $\mathcal{L}$. Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]
2012 USAMTS Round 1 p2
Three wooden equilateral triangles of side length $18$ inches are placed on axles as shown in the diagram to the right. Each axle is $30$ inches from the other two axles. A $144$-inch leather band is wrapped around the wooden triangles, and a dot at the top corner is painted as shown. The three triangles are then rotated at the same speed and the band rotates without slipping or stretching. Compute the length of the path that the dot travels before it returns to its initial position at the top corner.
A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.
2012 USAMTS Round 3 p3
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?
2013 USAMTS Round 2 p2
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
2013 USAMTS Round 3 p3
Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point.
Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.
2014 USAMTS Round 2 p3
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.
2014 USAMTS Round 3 p2
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
2015 USAMTS Round 3 p4
Let $\triangle ABC$ be a triangle with $AB<AC$. Let the angle bisector of $\angle BAC$ meet $BC$ at $D$, and let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $B$ to $\overline{AD}$. Extend $\overline{BP}$ to meet $\overline{AM}$ at $Q$. Show that $\overline{DQ}$ is parallel to $\overline{AB}$.
2016 USAMTS Round 1 p5
Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.
Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular $n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum possible value of $d$?
2016 USAMTS Round 3 p4
Let ${A_1, \dots , A_n }$ and ${B_1, \dots , B_n}$ be sets of points in the plane. Suppose that for all points $x$, $D \left( x , A_1 \right) + D \left( x , A_2 \right) + \cdots + D \left( x , A_n \right) \ge D \left( x , B_1 \right) + D \left( x , B_2 \right) + \cdots + D \left( x , B_n \right)$ where $D \left( x , y \right)$ denotes the distance between $x$ and $y$. Show that the $A_i$'s and the $B_i$'s share the same center of mass.
under construction
1998 USAMTS Round 1 p
1998 USAMTS Round 2 p
1998 USAMTS Round 3 p
1999 USAMTS Round 1 p
1999 USAMTS Round 2 p
1999 USAMTS Round 3 p
2000 USAMTS Round 1 p
2000 USAMTS Round 2 p
2000 USAMTS Round 3 p
2001 USAMTS Round 1 p
2001 USAMTS Round 2 p
2001 USAMTS Round 3 p
2002 USAMTS Round 1 p
2002 USAMTS Round 2 p
2002 USAMTS Round 3 p
2003 USAMTS Round 1 p
2003 USAMTS Round 2 p
2003 USAMTS Round 3 p
2004 USAMTS Round 1 p
2004 USAMTS Round 2 p
2004 USAMTS Round 3 p
2005 USAMTS Round 1 p
2005 USAMTS Round 2 p
2005 USAMTS Round 3 p
2006 USAMTS Round 1 p
2006 USAMTS Round 2 p
2006 USAMTS Round 3 p
2007 USAMTS Round 1 p
2007 USAMTS Round 2 p
2007 USAMTS Round 3 p
2008 USAMTS Round 1 p
2008 USAMTS Round 2 p
2008 USAMTS Round 3 p
2009 USAMTS Round 1 p
2009 USAMTS Round 2 p
2009 USAMTS Round 3 p
2010 USAMTS Round 1 p
2010 USAMTS Round 2 p
2010 USAMTS Round 3 p
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