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# NMTC 2023 Stage II - Kaprekar (Grade 7 & 8) - Problems and Solutions

###### Problem 1

If where and , prove that

###### Problem 2

are three distinct positive integers. Show that among the numbers there must be one which is divisible by 8 .

###### Problem 3

There are four points on a plane such that no three of them are collinear. Can the triangles and be such that at least one has an interior angle less than or equal to ? If so, how? If not, why?

###### Problem 4

A straight line is drawn through the vertex of an equilateral triangle , wholly lying outside the triangle. are drawn perpendiculars to the straight line . If is the midpoint of , prove that is an equilateral triangle.

###### Problem 5

is a parallelogram. Through , a straight line is drawn outside the parallelogram. and are drawn perpendicular to this line Show that . If the line through cuts one side internally, then will the same result hold? If so prove it. If not, what is the corresponding result? Justify your answer.

###### Problem 6

are non-negative real numbers whose sum is 1 . Prove that the maximum and minimum values of are respectively 1 and .

###### Problem 7

(a) Solve for

(b) If and , find the value of .

If , prove that

###### Problem 1

If where and , prove that

###### Problem 2

are three distinct positive integers. Show that among the numbers there must be one which is divisible by 8 .

###### Problem 3

There are four points on a plane such that no three of them are collinear. Can the triangles and be such that at least one has an interior angle less than or equal to ? If so, how? If not, why?

###### Problem 4

A straight line is drawn through the vertex of an equilateral triangle , wholly lying outside the triangle. are drawn perpendiculars to the straight line . If is the midpoint of , prove that is an equilateral triangle.

###### Problem 5

is a parallelogram. Through , a straight line is drawn outside the parallelogram. and are drawn perpendicular to this line Show that . If the line through cuts one side internally, then will the same result hold? If so prove it. If not, what is the corresponding result? Justify your answer.

###### Problem 6

are non-negative real numbers whose sum is 1 . Prove that the maximum and minimum values of are respectively 1 and .

###### Problem 7

(a) Solve for

(b) If and , find the value of .

###### Problem 8

If , prove that

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