geometry problems from Harvard-MIT Mathematics Tournament Invitational Competition (HMIC)
with aops links in the names
2013 HMIC p3
Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.
Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.
(a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides.
(b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.
$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?
2016 HMIC p2
Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other.
A polygon in the plane (with no self-intersections) is called equitable if every line passing through the origin divides the polygon into two (possibly disconnected) regions of equal area.
Does there exist an equitable polygon which is not centrally symmetric about the origin?
(A polygon is centrally symmetric about the origin if a $180$-degree rotation about the origin sends the polygon to itself.)
2020 HMIC p3
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
with aops links in the names
2013 - 2021
Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.
Allen Yuan
2014 HMIC p1Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.
(a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides.
(b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.
Anderson Wang
2014 HMIC p2$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?
2016 HMIC p2
Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other.
Dhroova Aiylam and Evan Chen
2018 HMIC p3A polygon in the plane (with no self-intersections) is called equitable if every line passing through the origin divides the polygon into two (possibly disconnected) regions of equal area.
Does there exist an equitable polygon which is not centrally symmetric about the origin?
(A polygon is centrally symmetric about the origin if a $180$-degree rotation about the origin sends the polygon to itself.)
2020 HMIC p3
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
Michael Ren
2020 HMIC p5
A triangle and a circle are in the same plane. Show that the area of the intersection of the triangle and the circle is at most one third of the area of the triangle plus one half of the area of the circle.
Krit Boonsiriseth
2021 HMIC p4
Let $A_1A_2A_3A_4$, $B_1B_2B_3B_4$, and $C_1C_2C_3C_4$ be three regular tetrahedra in $3$-dimensional space, no two of which are congruent. Suppose that, for each $i\in \{1,2,3,4\}$, $C_i$ is the midpoint of the line segment $A_iB_i$. Determine whether the four lines $A_1B_1$, $A_2B_2$, $A_3B_3$, and $A_4B_4$ must concur.
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