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

Problem 1

Find integers $m, n$ such that the sum of their cubes is equal to the square of their sum.

Problem 2

$PQR$ is an acute scalene triangle. The altitude $PL$ and the bisector $RK$ of $\angle QRP$ meet at $H$ ( $L$ on $QR$ and $K$ on $PQ$ ). $KM$ is the altitude of triangle $PKR$ ; it meets $PL$ at $N$ . The circumcircle of $\triangle NKR$ meets $QR$ at $S$ other than $Q$ . Prove that $SHK$ is an isosceles triangle.

Explore the Solution

Problem 3

Let $a_i(i=1,2,3,4,5,6)$ are reals. The polynomial

$f(x)=a_1+a_2 x+a_3 x^2+a_4 x^3+a_5 x^4+a_6 x^5+7 x^6-4 x^7+x^8$

can be factorized into linear factors $x-x_i$ where $i \in{1,2,3, \ldots, 8}$ .
Find the possible values of $a_1$ .

Problem 4

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 $2 n$ . Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$ .

Problem 5

$a, b, c$ are positive reals satisfying

$\frac{2}{5} \leq c \leq \operatorname{minimum}{a, b} ; a c \geq \frac{4}{15}$ and $b c \geq \frac{1}{5}$ . Find the maximum value of $\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)$

Problem 6

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.

Problem 7

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 $S(n)+9T(n) = n$ .

Problem 8

$ABCD$ is a cyclic quadrilateral. The midpoints of the diagonals $AC$ and $BD$ are respectively $P$ and $Q$ . If $BD$ bisects the $\angle AQC$ , then prove that $AC$ will bisect $\angle BPD$ .

Problem 1

Find integers $m, n$ such that the sum of their cubes is equal to the square of their sum.

Problem 2

$PQR$ is an acute scalene triangle. The altitude $PL$ and the bisector $RK$ of $\angle QRP$ meet at $H$ ( $L$ on $QR$ and $K$ on $PQ$ ). $KM$ is the altitude of triangle $PKR$ ; it meets $PL$ at $N$ . The circumcircle of $\triangle NKR$ meets $QR$ at $S$ other than $Q$ . Prove that $SHK$ is an isosceles triangle.

Explore the Solution

Problem 3

Let $a_i(i=1,2,3,4,5,6)$ are reals. The polynomial

$f(x)=a_1+a_2 x+a_3 x^2+a_4 x^3+a_5 x^4+a_6 x^5+7 x^6-4 x^7+x^8$

can be factorized into linear factors $x-x_i$ where $i \in{1,2,3, \ldots, 8}$ .
Find the possible values of $a_1$ .

Problem 4

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 $2 n$ . Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$ .

Problem 5

$a, b, c$ are positive reals satisfying

$\frac{2}{5} \leq c \leq \operatorname{minimum}{a, b} ; a c \geq \frac{4}{15}$ and $b c \geq \frac{1}{5}$ . Find the maximum value of $\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)$

Problem 6

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.

Problem 7

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 $S(n)+9T(n) = n$ .

Problem 8

$ABCD$ is a cyclic quadrilateral. The midpoints of the diagonals $AC$ and $BD$ are respectively $P$ and $Q$ . If $BD$ bisects the $\angle AQC$ , then prove that $AC$ will bisect $\angle BPD$ .

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