geometry problems from InfinityDots* Mathematical Olympiads
with aops links in the names
with aops links in the names
''A group of Thai students sharing the hobby of proposing olympiad math problems.
Most of us had been in the Thailand TST Camp, and several of us are IMO medalists.' quote source
aops posts collections:
2017 - 2019
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The line through $O$ parallel to $BC$ intersect $AB$ at $D$ and $AC$ at $E$. $X$ is the midpoint of $AH$. Prove that the circumcircles of $\triangle BDX$ and $\triangle CEX$ intersect again at a point on line $AO$.
Proposed by TacH
Given an acute triangle $\triangle ABC$ with circumcircle $\omega$ and circumcenter $O$.
The symmedian through $A$ intersects $\omega$ again at $S\neq A$.
Point $F$ is on $AC$ such that $BF\perp AS$, and point $G$ is on ray $BF$ such that $BF \times BG = BC^2$.
Finally, let $P$ be the point such that $\square BGCP$ is a parallelogram. Prove that $OS$ bisects $CP$.
Note: The symmedian is the reflection of the median over the internal angle bisector.
Proposed by talkon
2018 InfinityDots MO p3
Let $A,B,C$ be three distinct points on a line $\ell$. Prove that for each pair of distinct points $B_1,C_1$ such that $\overleftrightarrow{B_1C_1}$ does not pass through $A$, and $\overleftrightarrow{B_1C}$ is not parallel to $\overleftrightarrow{C_1B}$, there is a unique point $A_1$ satisfying:
(i) $A_1$ does not lie on $\overleftrightarrow{B_1C_1}$,
(ii) the projections of $A$ onto $\overleftrightarrow{B_1C_1}$, of $B$ onto $\overleftrightarrow{C_1A_1}$, and of $C$ onto $\overleftrightarrow{A_1B_1}$ lie on a line not parallel to $\ell$, and
(iii) the reflections of $A$ over $\overleftrightarrow{B_1C_1}$, of $B$ over $\overleftrightarrow{C_1A_1}$, and of $C$ over $\overleftrightarrow{A_1B_1}$ lie on a line not parallel to $\ell$.
Let $A,B,C$ be three distinct points on a line $\ell$. Prove that for each pair of distinct points $B_1,C_1$ such that $\overleftrightarrow{B_1C_1}$ does not pass through $A$, and $\overleftrightarrow{B_1C}$ is not parallel to $\overleftrightarrow{C_1B}$, there is a unique point $A_1$ satisfying:
(i) $A_1$ does not lie on $\overleftrightarrow{B_1C_1}$,
(ii) the projections of $A$ onto $\overleftrightarrow{B_1C_1}$, of $B$ onto $\overleftrightarrow{C_1A_1}$, and of $C$ onto $\overleftrightarrow{A_1B_1}$ lie on a line not parallel to $\ell$, and
(iii) the reflections of $A$ over $\overleftrightarrow{B_1C_1}$, of $B$ over $\overleftrightarrow{C_1A_1}$, and of $C$ over $\overleftrightarrow{A_1B_1}$ lie on a line not parallel to $\ell$.
Proposed by TacH
2019 InfinityDots MO p3
In a scalene triangle $ABC$, the incircle $\omega$ has center $I$ and touches side $BC$ at $D$. A circle $\Omega$ passes through $B$ and $C$ and intersects $\omega$ at two distinct points. The common tangents to $\omega$ and $\Omega$ intersect at $T$, and line $AT$ intersects $\Omega$ at two distinct points $K$ and $L$. Prove that either $KI$ bisects $\angle AKD$ or $LI$ bisects $\angle ALD$.
In a scalene triangle $ABC$, the incircle $\omega$ has center $I$ and touches side $BC$ at $D$. A circle $\Omega$ passes through $B$ and $C$ and intersects $\omega$ at two distinct points. The common tangents to $\omega$ and $\Omega$ intersect at $T$, and line $AT$ intersects $\Omega$ at two distinct points $K$ and $L$. Prove that either $KI$ bisects $\angle AKD$ or $LI$ bisects $\angle ALD$.
Proposed by talkon and ThE-dArK-lOrD
2019 InfinityDots MO p5
Is there a nonempty finite set $S$ of points on the plane that form at least $|S|^2$ harmonic quadrilaterals?
Note: a quadrilateral $ABCD$ is harmonic if it is cyclic and $AB\cdot CD = BC\cdot DA$.
Is there a nonempty finite set $S$ of points on the plane that form at least $|S|^2$ harmonic quadrilaterals?
Note: a quadrilateral $ABCD$ is harmonic if it is cyclic and $AB\cdot CD = BC\cdot DA$.
Proposed by talkon
2019 InfinityDots JMO p6
Determine all positive reals $r$ such that, for any triangle $ ABC$, we can choose points $D,E,F$ trisecting the perimeter of the triangle into three equal-length sections so that the area of $\triangle DEF$ is exactly $r$ times that of $\triangle ABC$.
Determine all positive reals $r$ such that, for any triangle $ ABC$, we can choose points $D,E,F$ trisecting the perimeter of the triangle into three equal-length sections so that the area of $\triangle DEF$ is exactly $r$ times that of $\triangle ABC$.
Proposed by talkon
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