USAMO 1972 - 2019 63p

geometry problems from United States of America Mathematical Olympiads (a.k.a USAMO)
with aops links in the names

USAMO 2000 - 19 EN with solutions by Evan Chen
bonus : USAMO 2003 Official Recommended Marking Scheme

more USA Competitions in appendix: UK USA Canada

1972 - 2019

A given tetrahedron $ABCD$ is isoceles, that is, $AB= CD$, $AC = BD$, $AD= BC$. Show that the faces of the tetrahedron are acute-angled triangles.

A given convex pentagon $ABCDE$ has the property that the area of each of five triangles $ABC, BCD, CDE, DEA$, and $EAB$ is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ < 60^\circ$.

Consider the two triangles $ABC$ and $PQR$ shown below. In triangle $ABC, \angle ADB = \angle BDC =\angle CDA= 120^\circ$. Prove that $x= u +v + w$.

Let $A,B,C,$ and $D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that $AC^2 + BD^2 + AD^2 BC^2 \ge AB^2 + CD^2.$

Two given circles intersect in two points $P$ and $Q$. Show how to construct a segment $AB$ passing through $P$ and terminating on the circles such that $AP \cdot PB$ is a maximum.

If $A$ and $B$ are fixed points on a given circle and $XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $AX$ and $BY$. You may assume that $AB$ is not a diameter.

If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\angle APB = \angle BPC=\angle CPA = 90^\circ$) is $S$, determine its maximum volume.

$ABC$ and $A'B'C'$ are two triangles in the same plane such that the lines $AA',BB',CC'$ are mutually parallel. Let $[ABC]$ denotes the area of triangle $ABC$ with an appropriate $\pm$ sign, etc.; prove that $3([ABC] +[A'B'C']) = [AB'C'] +[BC'A'] + [CA'B'] +[A'BC] + [B'CA] + [C'AB].$

Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.

$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.

a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?

Let $S$ be a great circle with pole $P$.  On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$.  For any  spherical triangle $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$.

Note: A great circle on a sphere is one whose center is the center of the sphere.  A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.

Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}+ \tfrac{1}{PC}$ is a maximum.
The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.

The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.

Note: A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.

1984 USAMO problem 3
$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.

Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.

Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.

Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.

$AD$, $BE$, and $CF$ are the bisectors of the interior angles of triangle $ABC$, with $D$, $E$, and $F$ lying on the perimeter.  If angle $EDF$ is $90$ degrees, determine all possible values of angle $BAC$.

Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$; $C_2$ is concentric and has diameter $k$ ($1 < k < 3$); $C_3$ has center $A$ and diameter $2k$.  We regard $k$ as fixed.  Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$, one endpoint $Y$ on $C_3$, and contain the point $B$.  For what ratio $XB/BY$ will the segment $XY$ have minimal length?

Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively.  Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.

Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.

An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.

Let $\, D \,$ be an arbitrary point on side $\, AB \,$ of a given triangle $\, ABC, \,$ and let $\, E \,$ be the interior point where $\, CD \,$ intersects the external common tangent to the incircles of triangles $\, ACD \,$ and $\, BCD$.  As $\, D \,$ assumes all positions between $\, A \,$ and $\, B \,$, prove that the point $\, E \,$ traces the arc of a circle.

Chords $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ of a sphere meet at an interior point $P$ but are not contained in a plane.  The sphere through $A$, $B$, $C$, $P$ is tangent to the sphere through $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $P$. Prove that $\, AA' = BB' = CC'$.

Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection.  Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.

A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $CP/PE = (AC/CE)^2$.

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

Let $ABC$ be a triangle.  Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.

Let ${\cal C}_1$ and ${\cal C}_2$ be  concentric circles, with ${\cal C}_2$ in the interior of  ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular  bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof,  the ratio $AM/MC$.

Let $ABCD$ be a cyclic quadrilateral. Prove that $|AB - CD| + |AD - BC| \geq 2|AC - BD|.$

Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$. The inscribed circle $\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$. Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove that the triangle $AFG$ is isosceles.

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$  where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$

2001 USAMO problem 2
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.

2001 USAMO problem 4
Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.

2002 USAMO problem 2
Let $ABC$ be a triangle such that
$\left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2,$
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

2003 USAMO problem 4
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that $\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.$  When does equality hold?

2004 USAMO problem 6
A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that $(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2.$ Prove that $ABCD$ is an isosceles trapezoid.

2005 USAMO problem 3
Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and $C_{1}$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and $B_{1}$ and $Q$ lie on opposite sides of line $AC$.  Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.

2006 USAMO problem 6
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

2007 USAMO problem 6
Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that
$8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; ,$ with equality if and only if triangle $ABC$ is equilateral.

2008 USAMO problem 2
Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Let the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.

2009 USAMO problem 1
Given circles $\omega_1$ and $\omega_2$ intersecting at points $X$ and $Y$, let $\ell_1$ be a line through the center of $\omega_1$ intersecting $\omega_2$ at points $P$ and $Q$ and let $\ell_2$ be a line through the center of $\omega_2$ intersecting $\omega_1$ at points $R$ and $S$.  Prove that if $P, Q, R$ and $S$ lie on a circle then the center of this circle lies on line $XY$.

2009 USAMO problem 5
Trapezoid $ABCD$, with $\overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $\omega$ and point $G$ lies inside triangle $BCD$.  Rays $AG$ and $BG$ meet $\omega$ again at points $P$ and $Q$, respectively.  Let the line through $G$ parallel to $\overline{AB}$ intersects $\overline{BD}$ and $\overline{BC}$ at points $R$ and $S$, respectively.  Prove that quadrilateral $PQRS$ is cyclic if and only if $\overline{BG}$ bisects $\angle CBD$.

2010 USAMO problem 1
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by
$P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively.  Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

2011 USAMO problem 3
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel.  The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$.  Furthermore $AB=DE$, $BC=EF$, and $CD=FA$.  Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.

2011 USAMO problem 5
Let $P$ be a given point inside quadrilateral $ABCD$.  Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.$ Prove that $\overline{Q_1Q_2}\parallel\overline{AB}$ if and only if   $\overline{Q_1Q_2}\parallel\overline{CD}$.

2012 USAMO problem 5
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

2013 USAMO problem 1
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively.  Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively.  Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.

2013 USAMO problem 6
Let $ABC$ be a triangle.  Find all points $P$ on segment $BC$ satisfying the following property:  If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then $\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.$

2014 USAMO problem 5
Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$.  Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$.  Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

2015 USAMO problem 2
Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$. As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.

2016 USAMO problem 3
Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively.  Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY=\angle CBY$ and $\overline{BE}\perp\overline{AC}$.  Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ=\angle BCZ$ and $\overline{CF}\perp\overline{AB}$.  Lines $\overleftrightarrow{I_BF}$ and $\overleftrightarrow{I_CE}$ meet at $P$.  Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.

by Evan Chen & Telv Cohl
2016 USAMO problem 5
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$. Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$.  Prove that $\overline{OI}$ is parallel to $\ell$, where $O$ is the circumcenter of triangle $ABC$, and $I$ is the incenter of triangle $ABC$.

2017 USAMO problem 3
Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$; the circle with diameter $\overline{DM}$ cuts $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $\overline{IS}$. The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L_2$. Prove that $\Omega$ passes through the midpoint of either $\overline{IL_1}$ or $\overline{IL_2}$.

by Evan Chen
2018 USAMO problem 5
In convex cyclic quadrilateral $ABCD$, we know that lines $AC$ and $BD$ intersect at $E$, lines $AB$ and $CD$ intersect at $F$, and lines $BC$ and $DA$ intersect at $G$. Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^\circ$.

2019 USAMO problem 2
Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$.

by Ankan Bhattacharya