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Stage 6 · Geometry

10 problems · National olympiad, first round · mathsolympiadprep.com

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1. Points $E$ and $F$ are the midpoints of edges $C C 1$ and $C 1 D 1$ of the rectangular parallelepiped $A B C D A 1 B 1 C 1 D 1$. The edge $K L$ of the regular triangular pyramid $K L M N$ (with $K$ as the vertex) lies on the line $A C$, and the vertices $N$ and $M$ lie on the lines $D D 1$ and $E F$ respectively. Find the ratio of the volumes of the prism and the pyramid, if $A B: B C=4: 3, K L: M N=2: 3$.

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2. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.

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   11. (20 points) Given the parabola $P: y^{2}=x$, with two moving points $A, B$ on it, the tangents at $A$ and $B$ intersect at point $C$. Let the circumcenter of $\triangle A B C$ be $D$. Is the circumcircle of $\triangle A B D$ (except for degenerate cases) always passing through a fixed point? If so, find the fixed point; if not, provide a counterexample.

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3. An isosceles triangle has a $108^{\circ}$ angle between its legs. Divide the triangle into the minimum number of acute-angled triangles.

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4. Let $ABC$ be a triangle with circumradius $R$, perimeter $P$ and area $K$. Determine the maximum value of: $\frac{KP}{R^3}$.

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5. On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?

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6. Let $T$ be a triangle with side lengths 3, 4, and 5. If $P$ is a point in or on $T$, what is the greatest possible sum of the distances from $P$ to each of the three sides of $T$?

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7. Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.

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8. Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $\text{xy 1}$ and both branches of the hyperbola $\text{xy 1.}$ (A set $S$ in the plane is called convex if for any two points in $S$ the line segment connecting them is contained in $S.$)

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9. Let $ABCD$ be a trapezoid such that $|AC|=8$, $|BD|=6$, and $AD \parallel BC$. Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|PS|=5$, find the area of the trapezoid $ABCD$.

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10. Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Evaluate the value of $\angle PSR$.

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Answer key — Stage 6 · Geometry

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

1. $\frac{25\sqrt{3}}{16}$ open
2. $(\frac{1}{4},0)$ open
3. $7$ open
4. $\frac{27}{4}$ open
5. $20$ open
6. $\frac{12}{5}$ open
7. $25$ open
8. $4$ open
9. $24$ open
10. $45^\circ$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.