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# NMTC 2023 Stage II Finals - Junior BHASKARA (Grade 9 & 10) - Problems and Solutions

###### Problem 1

Find integers such that the sum of their cubes is equal to the square of their sum.

###### Problem 2

is an acute scalene triangle. The altitude and the bisector of meet at ( on and on ). is the altitude of triangle ; it meets at . The circumcircle of meets at other than . Prove that is an isosceles triangle.

###### Problem 3

Let are reals. The polynomial

can be factorized into linear factors where .
Find the possible values of .

###### Problem 4

There are (an even number) bags. Each bag contains at least one apple and at most apples. The total number of apples is . Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is .

###### Problem 5

are positive reals satisfying

and . Find the maximum value of

###### Problem 6

The sum of the squares of four reals is 1 . Find the minimum value of the expression . Find also the values of and when this minimum occurs.

###### Problem 7

Let be a positive integer; and denote the sum of all digits in the decimal representation of . A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of is called a truncation of . The sum of all truncations of is denoted as . Prove that .

###### Problem 8

is a cyclic quadrilateral. The midpoints of the diagonals and are respectively and . If bisects the , then prove that will bisect .

###### Problem 1

Find integers such that the sum of their cubes is equal to the square of their sum.

###### Problem 2

is an acute scalene triangle. The altitude and the bisector of meet at ( on and on ). is the altitude of triangle ; it meets at . The circumcircle of meets at other than . Prove that is an isosceles triangle.

###### Problem 3

Let are reals. The polynomial

can be factorized into linear factors where .
Find the possible values of .

###### Problem 4

There are (an even number) bags. Each bag contains at least one apple and at most apples. The total number of apples is . Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is .

###### Problem 5

are positive reals satisfying

and . Find the maximum value of

###### Problem 6

The sum of the squares of four reals is 1 . Find the minimum value of the expression . Find also the values of and when this minimum occurs.

###### Problem 7

Let be a positive integer; and denote the sum of all digits in the decimal representation of . A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of is called a truncation of . The sum of all truncations of is denoted as . Prove that .

###### Problem 8

is a cyclic quadrilateral. The midpoints of the diagonals and are respectively and . If bisects the , then prove that will bisect .

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