Olympic geometry

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Problem: On the sides of the triangle \( A B C \) points \( A_{1} \in \) \( [B C], A_{2} \in\left[A_{1} C\right], B_{1} \in[C A], B_{2} \in\left[B_{1} A\right] \) are marked. \( C_{1} \in[A B], C_{2} \in\left[C_{1} B\right] \), for which \[ \begin{array}{l} \frac{C A_{1}}{B C}=\frac{C B_{2}}{C A}=\frac{B C+C A}{A B+B C+C A}, \\ \frac{A B_{1}}{C A}=\frac{A C_{2}}{A B}=\frac{C A+A B}{A B+B C+C A}, \\ \frac{B C_{1}}{A B}=\frac{B A_{2}}{B C}=\frac{A B+B C}{A B+B C+C A} . \end{array} \] Prove that the intersection points of the lines \( A_{1} C_{2}, C_{1} B_{2} \) and \( B_{1} A_{2} \) line on the circumscribed circle of the triangle \( A B C \).

12.1.1 Olympic geometry

9.51 $

Problem: In a trapezoid the ratio of the bases is \( 1: 3 \), and the diagonals 2:3. The lines, drawn through sides, are perpendicular. Find the ratio of the lengths of the sides.

12.1.2 Olympic geometry

2.85 $

Problem: Circle \( P \) is inscribed in the triangle TCS. Find the perimeter of the trapezoid, formed by the side \( T C \) and the line, passing through the centre \( P \) and the parallel \( T C \), if the bases of the trapezoid are equal to 3 and 5 .

12.1.4 Olympic geometry

3.57 $

Problem: On the side \( C D \) of the parallelogram \( A B C D \) points \( E_{1} \) and \( E_{2} \) are chosen so, that \( A B=B E_{1}=B E_{2} \). On the beam \( A E_{1} \) the point \( F_{1} \) is chosen so, that \( B E_{1} \| C F_{1} \), and on the beam \( A E_{2} \) the point \( F_{2} \) is chosen so, that \( B E_{2} \| C F_{2} \). Prove that \( D F_{1}=D F_{2} \).

12.1.5 Olympic geometry

3.57 $

Problem: Construct a section through the diagonal of the face of a cube, which is equal to the face of this cube.

12.1.6 Olympic geometry

4.04 $

Problem: On sides \( A B, B C, C D, D E, E F, F A \) of the regular hexagon \( A B C D E F \) with the area \( S \) points \( A_{1}, B_{1}, C_{1}, D_{1}, E_{1}, F_{1} \) are marked so, that \[ \frac{A A_{1}}{A_{1} B}=\frac{B B_{1}}{B_{1} C}=\frac{C C_{1}}{C_{1} D}=\frac{D D_{1}}{D_{1} E}=\frac{E E_{1}}{E_{1} F}=\frac{F F_{1}}{F_{1} A}=\frac{1}{4} . \]

12.1.7 Olympic geometry

4.76 $

Problem: What is the minimum height \( h \) of a box, having the shape of a rectangular parallelepiped with a length of 29 and a width of 27 that can fit two balls with radii \( R=12 \) and \( r=10 \) ?

12.1.8 Olympic geometry

4.76 $

Problem: What is the minimum height \( h \) of a box, having the form of a cuboid, with a length of 21 and a width of 18, which can fit two balls with radii \( R=9 \) and \( r=6 \) ?

12.1.3 Olympic geometry

4.76 $

Problem: On the side \( A C \) of the triangle \( A B C \) the point \( D \) is chosen. Circles, inscribed in triangles \( A B D \) and \( C B D \), are tangent to the segment \( B D \) at points \( P \) and \( Q \). What is the minimum natural value of \( x \) for which we can determine the relation \( A D: D C \) one-valued, if it is given that \( A B=21, B C=27, A C=30, P Q=x \) ?

12.1.9 Olympic geometry

2.85 $

Problem: The circle \( \Omega \) with radius \( R=1 \) and two lines, intersecting at the angle \( \alpha=30^{\circ} \), were constructed on the plane, moreover, one of these lines passes through the centre of the circle \( \Omega \), the other one is tangent to the circle \( \Omega \). How many different circles are there on the plane, each of which is tangent to the circle \( \Omega \) and both mentioned lines?

12.1.10 Olympic geometry

7.14 $

Problem: Cut a regular pentagon into five equal triangles and one smaller regular pentagon.

12.1.11 Olympic geometry

2.38 $

Problem: Square \( A B C D \) is inscribed in the circle \( \omega \). On the smaller \( \operatorname{arc} C D \) of the circle \( \omega \) an arbitrary point \( M \) is chosen. Inside the square such points as \( K \) and \( L \) are marked inside the square so, that \( K L M D \) is a square. Find \( \angle A K D \).

12.1.12 Olympic geometry

2.38 $