MathDB

Problem 3

Part of 1998 Croatia National Olympiad

Problems(4)

rate problem, find time spent

Source: Croatia 1998 1st Grade P3

6/7/2021
Ivan and Krešo started to travel from Crikvenica to Kraljevica, whose distance is 1515 km, and at the same time Marko started from Kraljevica to Crikvenica. Each of them can go either walking at a speed of 55 km/h, or by bicycle with the speed of 1515 km/h. Ivan started walking, and Krešo was driving a bicycle until meeting Marko. Then Krešo gave the bicycle to Marko and continued walking to Kraljevica, while Marko continued to Crikvenica by bicycle. When Marko met Ivan, he gave him the bicycle and continued on foot, so Ivan arrived at Kraljevica by bicycle. Find, for each of them, the time he spent in travel as well as the time spent in walking.
rateratesalgebra
triangle on midpoints of square sides

Source: Croatia 1998 2nd Grade P3

6/7/2021
Points EE and FF are chosen on the sides ABAB and BCBC respectively of a square ABCDABCD such that BE=BFBE=BF. Let BNBN be an altitude of the triangle BCEBCE. Prove that the triangle DNFDNF is right-angled.
geometrysquareTriangleright triangle
sum of altitude vectors is zero, then triangle is equilateral

Source: Croatia 1998 3rd Grade P3

6/8/2021
Let AA1,BB1,CC1AA_1,BB_1,CC_1 be the altitudes of a triangle ABCABC. If AA1→+BB1→+CC1→=0\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=0 prove that the triangle ABCABC is equilateral.
vectorgeometryTriangles
f(f(f(k)))=2n-k+1 over finite set

Source: Croatia 1998 4th Grade P3

6/8/2021
Let A={1,2,…,2n}A=\{1,2,\ldots,2n\} and let the function g:A→Ag:A\to A be defined by g(k)=2n−k+1g(k)=2n-k+1. Does there exist a function f:A→Af:A\to A such that f(k)≠g(k)f(k)\ne g(k) and f(f(f(k)))=g(k)f(f(f(k)))=g(k) for all k∈Ak\in A, if (a) n=999n=999; (b) n=1000n=1000?
combinatoricsfefunctional equation