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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.

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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
   $$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$$

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4. There is given a convex quadrilateral $ABCD$ and a point $P$ in the interior of the triangle $BCD$ in such a way that the quadrilateral $ABPD$ has an inscribed circle, and the three inscribed circles of the quadrilateral $ABPD$, the triangle $BCP$ and the triangle $CDP$, respectively, are pairwise tangent to each other. Denote by $Q$ and $R$ the points tangency on the line segments $BP$ and $DP$, respectively. Let the lines $BP$ and $AR$ meet at $S$, let the lines $DP$ and $AQ$ meet at $T$, and let the lines $BT$ and $DS$ meet at $U$. Show that the line $CU$ bisects the angle $BCD$.
   (5 pont)

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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 $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.

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

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7. In triangle $ABC$, we have $AB > AC$. The incircle $\omega$ touches $BC$ at $E$, and $AE$ intersects $\omega$ at $D$. Choose a point $F$ on $AE$ ($F$ is different from $E$), such that $CE = CF$. Let $G$ be the intersection point of $CF$ and $BD$. Prove that $CF = FG$.

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

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9. 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 $1$.

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10. Let $ABC$ be an acute-angled triangle with circumcenter $O$. Furthermore, let $k$ be a circle with the following properties:

    (1) The center $K$ of $k$ lies in the interior of the side $BC$.

    (2) $k$ touches $AB$ at $B'$ and $AC$ at $C'$.

    (3) $O$ lies on the shorter of the two arcs $B'C'$ of $k$.

    Prove: The circumcircle of $ABC$ and $k$ intersect each other in two distinct points.

    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. Prove it — see the worked solution 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

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