International Mathematical Olympiad 60', Second Day

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Sorry Everyone!, I was really busy the last whole week. Here is the next set of problems from the 1960 IMO.

Second Day

4. (HUN) Construct a triangle $ABC$ whose lengths of heights $h_a$ and $h_b$ (from A and B, respectively) and length of median $m_a$ (from A) are given.

5. (CZS) A cube $ABCDA'B'C'D'$ is given.

(a) Find the locus of all midpoints of segments $XY$ , where $X$ is any point on segment $AC$ and $Y$ any point on segment $B'D'$ .

(b) Find the locus of all points $Z$ on segments $XY$ such that $\vec{ZY} = 2 \vec{XZ}.$

6. (BUL) An isosceles trapezoid with bases $a$ and $b$ and height $h$ is given.

(a) On the line of symmetry construct the point P such that both (nonbase) sides are seen from P with an angle of $90^{\circ}$ .

(b) Find the distance of $P$ from one of the bases of the trapezoid.

(c) Under what conditions for $a, b$ , and $h$ can the point $P$ be constructed (analyze all possible cases)?

7. (GDR) A sphere is inscribed in a regular cone. Around the sphere a cylinder is circumscribed so that its base is in the same plane as the base of the cone. Let $V_1$ be the volume of the cone and $V_2$ the volume of the cylinder.

(a) Prove that $V1 = V2$ is impossible.

(b) Find the smallest $k$ for which $V_1 = k V_2$ , and in this case construct the angle at the vertex of the cone.

This is part of the set International Mathematical Olympiads

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2^{34}

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Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Xuming Liang @Nihar Mahajan here is the whole load of Olympiad Geometry! Enjoy!

Sualeh Asif - 5 years, 7 months ago

Sketch for a brutal solution for $4$ .

From $h_a,m_a$ we can construct the vertex $A$ , its foot of perpendicular on and midpoint of $BC$ , which are denoted $D,M$ respectively. From $h_a,h_b$ we know the ratio of $AC,BC$ and $MC,AC$ ; specifically $\frac {MC}{AC}=\frac {h_b}{2h_a}$ . Thus we can construct $C$ with the use of Apollonius circle, and $B$ consequently follows.

Xuming Liang - 5 years, 7 months ago

My solution to Q4:

Construction: Draw segment $AM$ with length $M_a$ . Now, draw a circle centred at $A$ with radius $h_a$ . Draw one of the tangents of this circle passing through $M$ . Now construct a circle with radius $\dfrac {h_b}{2}$ centred at $M$ . Draw one of the tangents of this second circle passing through $A$ . The intersection of these two tangents is $C$ . Draw a line parallel to the second tangent with a distance of $h_b$ between them. The intersection of this line and the firs tangent is $B$ . Thus, we have the 3 points $A$ , $B$ and $C$ to construct triangle $ABC$ .

Explanation: In a triangle $ABC$ , points $M$ and $N$ are the midpoints of sides $BC$ and $AC$ respectively. We have that $\Delta MNC \sim \Delta ABC$ and as a result point $M$ is a distance of $\dfrac {h_b}{2}$ away from $AC$ . The rest is self-explanatory.

Sharky Kesa - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$