Found problems: 7
2023 EGMO, 4
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Kvant 2023, M2746
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
2024 ELMO Shortlist, G6
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]
2024 ELMO Shortlist, G4
In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$.
[i]Neal Yan[/i]
2024 ELMO Shortlist, G6
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]
2024 ELMO Shortlist, G4
In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$.
[i]Neal Yan[/i]
2024 ELMO Problems, 5
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]