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Stage 6 · Geometry

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

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

1. Given the triangle $ABC$ right-angled at $A$, we construct on the hypotenuse the square $BCDE$ (with $D, E$ on the opposite side of $A$ with respect to $BC$). Knowing that the areas of triangles $ABE$ and $ACD$ are respectively $6~\mathrm{m}^2$ and $27~\mathrm{m}^2$, what is the area of triangle $ABC$?

   1. A$3 \sqrt{2}~m^2$
   2. B$6~\mathrm{m}^2$
   3. C$12~\mathrm{m}^2$
   4. D$9 \sqrt{2}~\mathrm{m}^2$
   5. E$18~\mathrm{m}^2$

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2. A rectangle with side lengths $1$ and $3$, a square with side length $1$, and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
   

   1. A14
   2. B23
   3. C46
   4. D59
   5. E67

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3. 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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4. (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.

   保留源文本的换行和格式如下：

   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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5. 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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6. 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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7. 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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8. 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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9. 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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10. In the cyclic quadrilateral $ABCD$, the diagonal $AC$ bisects the angle $DAB$. The side $AD$ is extended beyond $D$ to a point $E$. Show that $CE = CA$ if and only if $DE = AB$.

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Answer key — Stage 6 · Geometry

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

1. $D$ open
2. $E$ open
3. $\frac{25\sqrt{3}}{16}$ open
4. $(\frac{1}{4},0)$ open
5. $7$ open
6. $\frac{27}{4}$ open
7. $20$ open
8. $\frac{12}{5}$ open
9. $25$ open
10. Prove it — see the worked solution 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.