For teachers / Printable sets /Stage 8 · Geometry

Stage 8 · Geometry

10 problems · IMO Shortlist mid-range; USAMO P2/P5 · mathsolympiadprep.com

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

1. Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.

   Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.

   Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.

   Geometry Solution and answer checking →

2. Given positive integer $n \ge 5$ and a convex polygon $P$, namely $A_1A_2...A_n$. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $A_iA_jA_kA_l (1\le i<j<k<l\le n)$ not on diagonals of $P$, such that the $\tbinom{n}{4}$ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

   Geometry Solution and answer checking →

3. Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that
   $$\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.$$
   Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum
   $$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$$

   Geometry Solution and answer checking →

4. Let $n \geq 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$, with $A$ to the left of $B$, and letting the flea from $A$ jump over the flea from $B$ to the point $C$ so that $\text{BC AB}$.

   Determine all values of $\lambda$ such that, for any point $M$ on the line and for any initial position of the $n$ fleas, there exists a sequence of moves that will take them all to the position right of $M$.

   Geometry Solution and answer checking →

5. In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$. The x-intercept of line $A_nB_n$ is $a_n$, and the x-coordinate of point $B_n$ is $b_n$, $n\in\mathbb{N}$. Prove that
   (1) $a_n>a_{n+1}>4$, $n\in\mathbb{N}$;
   (2) There is $n_0\in\mathbb{N}$, such that for any $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.

   Geometry Solution and answer checking →

6. Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect in a point $P$. Prove that
   $$\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}$$

   Geometry Solution and answer checking →

7. A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.
   .

   Geometry Solution and answer checking →

8. a) In a triangle $MNP$, the lenghts of the sides are less than $2$. Prove that the lenght of the altitude corresponding to the side $MN$ is less than $\text{4 MN 2 4}$.
   
   b) In a tetrahedron $ABCD$, at least $5$ edges have their lenghts less than $2$.Prove that the volume of the tetrahedron is less than $1$.

   Geometry Solution and answer checking →

9. Given are three pairwise externally tangent circles $K_{1}$ , $K_{2}$ and $K_{3}$. denote by $P_{1}$ tangent point of $K_{2}$ and $K_{3}$ and by $P_{2}$ tangent point of $K_{1}$ and $K_{3}$.
   
   Let $AB$ ($A$ and $B$ are different from tangency points) be a diameter of circle $K_{3}$. Line $AP_{2}$ intersects circle $K_{1}$ (for second time) at point $X$ and line $BP_{1}$ intersects circle $K_{2}$(for second time) at $Y$.
   
   If $Z$ is intersection point of lines $AP_{1}$ and $BP_{2}$ prove that points $X$, $Y$ and $Z$ are collinear.

   Geometry Solution and answer checking →

10. Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$, where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line, perpendicular to $BD$ at $E$, intersects $CD$ at $M$, prove that $KM \perp DL$.

    Geometry Solution and answer checking →

Answer key — Stage 8 · Geometry

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

1. $K \text{ is the intersection of } AB' \text{ and } BA', \text{ and } c \text{ is a constant}$ open
2. $\text{Proven}$ open
3. $4$ open
4. $\lambda \ge \frac{1}{n-1}$ open
5. Prove it — see the worked solution open
6. Prove it — see the worked solution open
7. Prove it — see the worked solution open
8. Prove it — see the worked solution open
9. Prove it — see the worked solution 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.