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Find integers m, n such that the sum of their cubes is equal to the square of their sum.
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Find integers m, n such that the sum of their cubes is equal to the square of their sum.
PQR is an acute scalene triangle. The altitude PL and the bisector RK of $$\angle QRP$$ meet at H, with L on QR and K on PQ. KM is the altitude of triangle PKR and it meets PL at N. The circumcircle of $$\triangle NKR$$ meets QR at S other than R. Prove that SHK is an isosceles triangle.
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Let $$a_i$$ for $$i = 1, 2, 3, 4, 5, 6$$ be reals. The polynomial $$f(x) = a_1 + a_2x + a_3x^2 + a_4x^3 + a_5x^4 + a_6x^5 + 7x^6 - 4x^7 + x^8$$ can be factorized into linear factors $$x - x_i$$ where $$i = 1, 2, 3, \ldots, 8$$. Find the possible values of $$a_1$$.
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There are n, an even number, bags. Each bag contains at least one apple and at most n apples. The total number of apples is 2n. Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is n.
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a, b, c are positive reals satisfying $$\frac{2}{5} \leq c \leq \min(a, b)$$, $$ac \geq \frac{4}{15}$$ and $$bc \geq \frac{1}{5}$$. Find the maximum value of $$\frac{1}{a} + \frac{2}{b} + \frac{3}{c}$$.
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The sum of the squares of four reals x, y, z, u is 1. Find the minimum value of the expression $$E = (x-y)(y-z)(z-u)(u-x)$$. Find also the values of x, y, z and u when this minimum occurs.
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Let n be a positive integer and $$S(n)$$ denote the sum of all digits in the decimal representation of n. A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of n is called a truncation of n. The sum of all truncations of n is denoted as $$T(n)$$. Prove that $$S(n) + 9T(n) = n$$.
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ABCD is a cyclic quadrilateral. The midpoints of the diagonals AC and BD are respectively P and Q. If BD bisects $$\angle AQC$$, then prove that AC will bisect $$\angle BPD$$.
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