The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 31), and (31, 0) is

Determine the greatest number among the following four numbers

Two small insects, which are x metres apart, take u minutes to pass each other when they are flying towards each other, and v minutes to meet each other when they are flying in the same direction. Then, the ratio of the speed of the slower insect to that of the faster insect is

Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 \(cm^2\). Then the possible values of the sum of their volumes are________

An alloy P has copper and zinc in the proportion of 5: 2 (by weight), while another alloy Q has the same metals in the proportion of 3: 4 (by weight). If these two alloys are mixed in the proportion of a : b (by weight), a new alloy R is formed, which has equal contents of copper…

The inequality…

How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000 ?

The value of \((\log_3 30)^{-1}+(\log_4 900)^{-1}+(\log_5 30)^{-1}\) is

There are numbers \(a_1,a_2,a_3,\ldots,a_n\) each of them being +1 or -1. If it is known that \(a_1a_2+a_2a_3+a_3a_4+\cdots+a_{n-1}a_n+a_na_1=0\) then