Find number of 8 digit numbers sum of whose digits is 4. Discussion

Find the number of 4-tuples (a,b,c,d) of natural numbers with and Discussion

In an acute angled triangle ABC with AB < AC the circle touches AB at B and passes through C intersecting AC again at D. Prove that the orthocenter of triangle ABD lies on if and only if it lies on the perpendicular bisector of BC.

A polynomial is called a Fermat Polynomial if it can be written as the sum of squares of two polynomials with integer coefficients. Suppose f(x) is a Fermat Polynomial such that f(0) = 1000. Show that f(x) + 2x is not a Fermat Polynomial.

Let ABC be a triangle which is not right angled. Define a sequence of triangles with \( i \ge 0\) as follows. and for are the reflections of the orthocenter of triangle in the sides respectively. Assume that for some distinct natural numbers m, n. Prove that .

Let be a natural number. Let be a regular polygon and X = { 1, 2, …, n }. A subset , , \( i_1 < i_2 < … < i_k \) is called a good subset if the angles of the polygon angles \( A_{i_1} … A_{i_k}\) when arranged in an increasing order is an arithmetic progression. If n is prime then show that a PROPER good subset of X contains exactly 4 elements.

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