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$$l, m, n$$ are positive integers such that $$m$$ divides $$l+1$$, $$n$$ divides $$m+1$$ and $$l$$ divides $$n+1$$. Find all possible triples $$(l, m, n)$$.
Consider 43 positive integers. Show that there exist two of them, say x and y, such that $$x^2 - y^2$$ is divisible by 100.
ABCD is a quadrilateral and G, H are the middle points of AC and DB respectively. If CB, DA are produced to meet at E, prove that the area of $$\triangle EGH$$ is equal to $$\frac{1}{4}$$ (area of quadrilateral ABCD).
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a, b, c are three non-negative real numbers whose sum is not greater than $$\frac{1}{2}$$. Show that $$(1-a)(1-b)(1-c) \ge \frac{1}{2}$$.
The vertical angle A of an isosceles triangle ABC is $$\frac{1}{3}$$ of each of the base angles. Two points M and N are taken on AB and AC respectively so that $$BM = BC = CN$$. If BN and CM cut at P, show that $$\angle MPN = \angle B$$.
Solve for x, y, z.
$$12(x+y) = 5xy$$,
$$24(y+z) = 7yz$$,
$$8(z+x) = 3zx$$.
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Ten line segments are given. Each line segment has length between 1 cm and 55 cm. Prove that you can select three line segments such that they form a triangle.
ABC and DAC are two isosceles triangles, with $$\angle BAC = 20^\circ$$ and $$\angle ADC = 100^\circ$$. Prove that $$AB = BC + CD$$.
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