Olympiad problems

2023 NMTC Junior, P7

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 the [i]truncation[/i] of $n$. The sum of all truncations of $n$ is denoted as $T(n)$. Prove that $S(n)+9T(n)=n$

2023 NMTC Junior, P8

Tags: geometry , nmtc , cyclic quadrilateral

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

2023 NMTC Junior, P4

Tags: nmtc , combinatorics

There are $n$ (an even number) bags. Each bag contains atleast one apple and at most $n$ apples. The total number of apples is $2n$. Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$.

2023 NMTC Junior, P3

Tags: algebra , polynomial , nmtc

Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial $f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where $i \in {1,2,3,...,8}$. Find the possible values of $a_1$.

2023 NMTC Junior, P5

Tags: inequalities , nmtc , algebra

$a,b,c$ are positive reals satisfying $\frac{2}{5} \leq c \leq \min{a,b}$ ; $ac \geq \frac{4}{15}$ and $bc \geq \frac{1}{5}$ Find the maximum value of $\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)$.

2023 NMTC Junior, P2

Tags: geometry , circumcircle , nmtc

$PQR$ 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.

2023 NMTC Junior, P6

Tags: inequalities , nmtc

The sum of 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 minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs.

2023 NMTC Junior, P1

Tags: number theory , nmtc

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