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x, y, z are three consecutive integers. If $$\frac{x^3+y^3+z^3}{xyz}$$ is also an integer, find all possible values of $$(x, y, z)$$.
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x, y, z are three consecutive integers. If $$\frac{x^3+y^3+z^3}{xyz}$$ is also an integer, find all possible values of $$(x, y, z)$$.
ABC is an equilateral triangle. $$AB=10\text{cm}$$. D is a point outside $$\triangle ABC$$ such that $$BD=DC$$ and $$\angle BDC = 120^\circ$$. Points M, N are on sides AB and AC respectively such that $$\angle MDN = 60^\circ$$. Find the perimeter of $$\triangle AMN$$.
Find the value of $$\frac{2025^4+2026^4+1}{2025^2+2026^2+1}$$.
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$$(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$$ is a set of seven positive integers. Find the number of such 7-element sets of positive integers if $$a_1 \times a_2 \times a_3 = 70$$, $$a_3 \times a_4 \times a_5 = 71$$, and $$a_5 \times a_6 \times a_7 = 72$$.
Solve for x. $$\frac{x}{24} + \frac{x}{104} + \frac{x}{234} + \frac{x}{414} + \frac{x}{644} + \frac{x}{924} + \frac{x}{1254} = 49$$.
In a classroom there are 25 students. Their teacher writes 1 at both ends of the blackboard. The first student adds a 2 in the middle between them; each next student adds the sum of each two adjacent numbers already on the blackboard between them. Hence there are numbers 1, 2, 1 after the 1st student; 1, 3, 2, 3, 1 after the second student; 1, 4, 3, 5, 2, 5, 3, 4, 1 after the third student and so on. Find the sum of all numbers on the blackboard after the twenty-fifth student.
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In $$\triangle YRL$$, $$YR=RL$$. Equilateral $$\triangle YRV$$ and $$\triangle RLS$$ are drawn on the sides YR and RL respectively in the exterior of $$\triangle YRL$$. Line YS and line LV intersect in M, then show that $$MV=MS$$.
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