A fun run has a choice of two routes as shown below.The long route is around the whole perimeter of the shape 𝐴⁢𝐵⁢𝐶⁢𝐷, starting and finishing at 𝐴.The short route is around the rectangle 𝐴⁢𝐵⁢𝐸⁢𝐹, starting and finishing at 𝐴.

A, B, C, D, and E practice running at a stadium that has several tracks. Each one starts running from the same point Z. A runs around a path which is an equilateral triangle ZYX. B runs around a square path ZYWV. C runs on a regular pentagonal path ZYUTS. D runs on a regular hexagonal path ZYRQPO. E runs along the trapezium ZYPO and completes one round in 20 seconds. Among all the tracks, there is a common side ZY, having length 100 m. The ratio of speeds of B and C is 5 : 4 respectively whereas the ratio of the time taken by A and D to complete one round of their tracks is 3 : 4 respectively. E's speed is the same as B's speed whereas A's speed is half that of C's.Note: The athletes maintain the same speeds in any track.Q 113.   If A, B and C start running in the same direction along a 300 m long circular track, then after how much time will they meet for the first time?a)  60 secondsb)  90 secondsc)  75 secondsd)  120 seconds

A runs 4 times as fast as B. If A gives B a start of 60 m, how far must the goal on the race course be so that A and B reach it at the same time?

A race consists of three tracks – an uphill track from A to B, a downhill track from B to C, and an uphill track from C to D. The lengths of the three tracks are equal. Anil travelled from A to D covering the uphill and downhill tracks at 6 km/hr and 9 km/hr respectively. He took hours to reach D. Find the time that he would take to return from D to A (in hrs).2

There is rectangle of size 𝑁×𝑀N×M with opposite corners at (0,0)(0,0) and (𝑁,𝑀)(N,M); and a special point (𝑥+0.5,𝑦+0.5) (0≤𝑥<𝑁,0≤𝑦<𝑀)(x+0.5,y+0.5) (0≤x<N,0≤y<M).Two players play a game on the rectangle where each player takes alternate turns. In his/her turn, the player will choose a line, either 𝑥=𝑘x=k or 𝑦=𝑘y=k, such that:𝑘k is an integer;The chosen line divides the current rectangle into two non-empty parts.The part of rectangle that does not consist of the special point, is discarded for further moves.The player who cannot make a move loses. If both players play optimally, determine the number of special points such that the first player wins.Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case contains two space-separated integers 𝑁N and 𝑀M — the lengths of the sides of the rectangle.Output FormatFor each test case, print the number of special points (𝑥+0.5,𝑦+0.5)(x+0.5,y+0.5) for which the first player wins the game.Note that 0≤𝑥<𝑁0≤x<N and 0≤𝑦<𝑀0≤y<M.Constraints1≤𝑇≤1041≤T≤10 4 1≤𝑁,𝑀≤1061≤N,M≤10 6 The sum of 𝑁N as well as the sum of 𝑀M over all test cases does not exceed 10610 6 .Sample 1:InputOutput42 12 23 51 120100Explanation:Test case 11 : There are 22 possible special points (0.5,0.5)(0.5,0.5) and (1.5,0.5)(1.5,0.5). In both cases, the first player can select the line 𝑥=1x=1 on his first move, and then the second player is left with no moves. Thus, the first player wins for both the special points.Test case 22 : There are 44 possible special points, all of them end up losing for the first player.

Two circular paths P1 and P2 of radii 150 m and 50 m, respectively touch at a point B. Starting from B at the same time, Alex and Andy are walking on path P1 and path P2 at speeds 20 km/hr and 10 km/hr, respectively. The number of full rounds that Alex will make before he meets Andy again for the first time is: