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Stage 5 · Geometry

10 problems · AIME late · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Graph theory's Four-Color Theorem says that four colors are enough to color the regions in a plane so that no two adjacent regions receive the same color. The theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken, 124 years after the Four-Color Problem was posed.

   Fermat's Last Theorem in Number Theory was proved by Andrew Wiles in 1995, after 358 years of attempts by generations of mathematicians.

   In 2003, Grigori Perelman completed the proof of a conjecture in topology. Considered as one of the seven millennium prize problems, the conjecture says that the sphere is the only type of bounded three-dimensional surface that contains no holes. Mathematicians worked on this conjecture for almost a century. What is the name of this conjecture that earned Perelman the Fields Medal which he refused to accept in 2006?

   Tenth Philippine Mathematical Olympiad

   Geometry Solution and answer checking →

2. Let $ABCD$ be a quadrilateral such that $AB=24$, $BC=20$, $CD=15$, $DA=7$, $BD=25$. How long is $AC$?

   1. A18
   2. B$14\sqrt{2}$
   3. C20
   4. D21
   5. E24

   Progetto Olimpiadi della Matematica

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3. A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $CD=CE$, compute $\lfloor r \rfloor$.

   HMMT February

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4. Let $ABCD$ be a cyclic quadrilateral, and let segments $AC$ and $BD$ intersect at $E$. Let $W$ and $Y$ be the feet of the altitudes from $E$ to sides $DA$ and $BC$, respectively, and let $X$ and $Z$ be the midpoints of sides $AB$ and $CD$, respectively. Given that the area of $AED$ is $9$, the area of $BEC$ is $25$, and $\angle EBC - \angle ECB = 30^{\circ}$, then compute the area of $WXYZ$.

   HMMT February 2020

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5. Suppose that $ABC$ is an isosceles triangle with $AB=AC$. Let $P$ be the point on side $AC$ so that $AP=2CP$. Given that $BP=1$, determine the maximum possible area of $ABC$.

   HMMT February

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6. Let $A B C$ be a triangle with incircle tangent to the perpendicular bisector of $B C$. If $B C=A E=$ 20, where $E$ is the point where the $A$-excircle touches $B C$, then compute the area of $\triangle A B C$.

   HMMT February

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7. Let $S$ be the set of lattice points inside the circle $x^{2}+y^{2}=11$. Let $M$ be the greatest area of any triangle with vertices in $S$. How many triangles with vertices in $S$ have area $M$?

   HMMT February

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8. Suppose $\triangle A B C$ has lengths $A B=5, B C=8$, and $C A=7$, and let $\omega$ be the circumcircle of $\triangle A B C$. Let $X$ be the second intersection of the external angle bisector of $\angle B$ with $\omega$, and let $Y$ be the foot of the perpendicular from $X$ to $B C$. Find the length of $Y C$.

   HMMT February

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9. Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.

   HMMT February

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10. Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of the extensions of $\overline{A B}$ and $\overline{A C}$ with $l$. Compute $F G$.

    HMMT February

    Geometry Solution and answer checking →

Answer key — Stage 5 · Geometry

Worked solutions for every problem are on the site, one page per problem.

1. $Poincaré Conjecture$ open
2. $C$ open
3. $673$ open
4. $17 + (15/2)*sqrt(3)$ open
5. $\frac{9}{10}$ open
6. $100 \sqrt{2}$ open
7. $16$ open
8. $\frac{13}{2}$ open
9. $\frac{7}{2}$ open
10. $\frac{1024}{45}$ open

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