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The pigeonhole principle
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AABC ABCD acute angle af(x altitudes arithmetic mean At+i Chebyshev Polynomial coefficients Competition contains convex denote desired locus diameter difference digit of a2 distance divides divisible endpoints equation Figure follows function geometric mean given circle given integers greater greatest common divisor harmonic mean Hence hypotenuse identity inequality inscribed inside integers intersection Jensen's theorem KLMN lengths less than 180 mathematician midpoints multiple natural numbers nomogram Note obtain obtuse odd number pair parallelogram perpendiculars to BC plane polygon polynomials positive integer positive numbers powers with respect prime factorization problem proof Prove Pythagoras quadratic quadrilateral quadrilateral A BCD radius rational number real numbers relatively prime rhombus right triangle satisfy Second Solution Similarly sin4 smallest exponent square straight line Suppose tangent term triangle ABC trigonometric polynomial values vertex vertices xi xi yields zero