MathDB

9.4

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

x^2- 10[x] + 9 = 0 (II Soros Olympiad 1995-96 R1 9.4)

Source:

6/2/2024
Solve the equation x2−10[x]+9=0x^2- 10[x] + 9 = 0.
([x][x] is the integer part of xx, [x][x] is equal to the largest integer not exceeding xx. For example, [3,33]=3[3,33] = 3, [2]=2[2] = 2, [−3.01]=−4[- 3.01] = -4).
algebrafloor function
100 schoolchildren, 4 tasks (II Soros Olympiad 1995-96 R2 9.4)

Source:

6/3/2024
100100 schoolchildren took part in the Mathematical Olympiad. 44 tasks were proposed. The first problem was solved by 9090 people, the second by 8080, the third by 7070 and the fourth by 6060. However, no one solved all the problems. Students who solved both the third and fourth questions received an award. How many students were awarded?
number theoryalgebra
colored lines x = k, y = m, 19x+96y= c

Source: II Soros Olympiad 1995-96 R3 9.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/4/2024
All possible vertical lines x=kx = k and horizontal lines y=my = m are drawn on the coordinate plane, where kk and mm are integers. Let's imagine that all these straight lines are black. A red straight line is also drawn, the equation of which is 19x+96y=c19x+96y= c. Let us denote by MM the number of segments of different lengths formed on the red line when intersecting with the black ones.(The ends of each segment are the intersection points of the red and black lines. There are no such intersection points inside the segment.) What values can MM take when cc changes?
combinatoricscombinatorial geometryColoring