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Let ABC be a non-equilateral triangle with integer sides. Let D and E be respectively the mid-points of BC and AC; let G be the centroid of ABC. Suppose D ...

If \(x^{2}-10ax -11b=0\) has roots \(c\) and \(d\) and \(x^{2}-10cx-11d = 0\) has roots \(a\) and \(b\), then find the value of \(a+b+c+d\).

Let 2x+3y+ 5z+7k+9w be divisible by 4, where x, y, z, k and w are digits. If the number of solutions of ( x, y, z, k, w ...

Determine the number of ordered pairs of positive integers \((a, b)\) such that the least common multiple of \(a\) and \(b\) is \(2^{3} 5^{7} 11^{13}\).

Find all distinct positive integers p, q, r, s > 1 such that p! + q! + r! = 2^s.

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