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Stage 7 · Geometry

10 problems · National olympiad second round; IMO P1/P4 · mathsolympiadprep.com

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

1. Given triangle $ABC$, let $M$ be the midpoint of side $AB$ and $N$ be the midpoint of side $AC$. A circle is inscribed inside quadrilateral $NMBC$, tangent to all four sides, and that circle touches $MN$ at point $X.$ The circle inscribed in triangle $AMN$ touches $MN$ at point $Y$, with $Y$ between $X$ and $N$. If $XY=1$ and $BC=12$, find, with proof, the lengths of the sides $AB$ and $AC$.

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2. When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

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3. The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?
   

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4. A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$:

   1. A12
   2. B13
   3. C14
   4. D15
   5. E16

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5. Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?

   1. A59
   2. B60
   3. C61
   4. D62
   5. E63

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6. Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon.

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7. Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

   Note that this problem is null because a diagram is impossible.

   Proposed by Andy Xu

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8. Let $\Gamma$ be a circle centered at $O$ with chord $AB$. The tangents to $\Gamma$ at $A$ and $B$ meet at $C$. A secant from $C$ intersects chord $AB$ at $D$ and $\Gamma$ at $E$ such that $D$ lies on segment $CE$. Given that $\angle BOD + \angle EAD = 180^\circ$, $AE = 1$, and $BE = 2$, find $CE$.

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9. Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
   balls?

   1. A7
   2. B9
   3. C11
   4. D13
   5. E15

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10. Triangle $A_1 A_2 A_3$ has a right angle at $A_3$. A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ($n \geq 3$), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$.
    (a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$, $n \geq 3$.
    (b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$.

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Answer key — Stage 7 · Geometry

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

1. $AB = 12$ open
2. $\frac{7 - 3\sqrt{5}}{2}$ open
3. $100\pi - 40 \text{ kr.}$ open
4. $13$ open
5. $59$ open
6. $n = 6$ open
7. $622$ open
8. $\frac{4\sqrt{2}}{3}$ open
9. $13$ open
10. $\text{Locus of } P \text{ is an arc of the circle with diameter } A_1A_3 \text{ subtending an angle of } 2\arctan \frac{1}{2} \text{ at } A_1, \text{ excluding } A_3.$ open

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