**B. 4422.** There are 99 sticks lying on a table, their lengths are 1,2,3,...,99 units. Andrea and Bill play the following game: they take turns removing one stick of their choice. Andrea starts the game. The game ends when there are exactly three sticks remaining on the table. If it is possible to make a triangle out of the three sticks then Andrea wins. Otherwise, Bill is the winner. Who has a winning strategy?

(5 points)

**B. 4429.** *A*_{1}*B*_{1}*C*_{1} and *A*_{2}*B*_{2}*C*_{2} are two triangles such that their sides *A*_{1}*B*_{1} and *A*_{2}*B*_{2}, *B*_{1}*C*_{1} and *B*_{2}*C*_{2}, as well as *A*_{1}*C*_{1} and *A*_{2}*C*_{2} are parallel. Vertex *A*_{1} is connected to vertices *B*_{2} and *C*_{2}, vertex *B*_{1} is connected to *C*_{2} and *A*_{2}, and vertex *C*_{1} is connected to *A*_{2} and *B*_{2}. Given that the areas of the original triangles are *T*_{1} and *T*_{2}, what may be the area of the hexagon formed by the midpoints of the connecting line segments obtained in this way?

(4 points)

**C. 1110.** Andrew is going for a walk, starting from a corner of a block in a housing estate where streets form a rectangular lattice. During his walk, he only changes direction at street corners. The buildings form square blocks, 15 m on a side, and the width of the streets is negligible. Show that if Andrew ends his walk at the starting point, then the length of his path in metres will be an even number.

(5 points)

**K. 326.** An interior designer is planning the illumination of a large lecture hall. He is using LEDs (light emitting diodes) arranged in concentric circles. The LEDs are uniformly spaced along each circle. The radius of each circle is the double of the previous circle. If lines are drawn from each lead of a circle through the centre, only every other line will have a LED on the next circle inside. *a*) Show that the separation of two consecutive LEDs on the same circle, as measured along the circle, is constant (that is, independent of the circle selected). *b*) Determine this distance (measured along the circle) if the radius of the largest circle is 20 metres, the number of circles is 8, and the 4th smallest circle contains 112 LEDs. *c*) What is the total number of LEDs used?

(6 points)

This problem is for grade 9 students only.

**K. 327.** The sum of four positive integers is 125. If the first number is increased by 4, the second number is decreased by 4, the third number is multiplied by 4 and the fourth number is divided by 4, the results will be all equal. What may be the four original numbers?

(6 points)

This problem is for grade 9 students only.

**K. 328.** 0, 1, 2, 3 are substituted for *a*, *b*, *c*, *d* in the expression *a*^{b}+*cd* in all possible orders. What will be the sum of the resulting numbers?

(6 points)

This problem is for grade 9 students only.