10.5
Problems(3)
king in 7x7 chessboard (IV Soros Olympiad 1997-98 Correspondence 10.5)
Source:
6/1/2024
In the lower left corner of the square board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?
combinatoricscombinatorial geometry
triangle connstruction, 3 rays with common origin
Source: IV Soros Olympiad 1997-98 R2 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
6/1/2024
Three rays with a common origin are drawn on the plane, dividing the plane into three angles. One point is marked inside each corner. Using one ruler, construct a triangle whose vertices lie on the given rays and whose sides contain the given points.
geometryconstruction
all plane angles at vertex A are equal, triangular pyramid
Source: IV Soros Olympiad 1997-98 R3 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics
6/2/2024
At the base of the triangular pyramid lies a regular triangle such that . All plane angles at vertex are equal to each other. What might these angles be equal to?
geometryangle bisector3D geometrypyramid