2023 NMTC Junior

Part of NMTC

Subcontests

(8)

1

NMTC Junior 2023 P8

ABCD

is a cyclic quadrilateral. The midpoints of the diagonals

AC

BD

are respectively

P

Q

BD

\angle AQC

, the prove that

AC

\angle BPD

1

NMTC Junior 2023 P7

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 truncation of

n

. The sum of all truncations of

n

T(n)

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

1

NMTC Junior 2023 P6

The sum of squares of four reals

x,y,z,u

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

u

when this minimum occurs.

1

NMTC 2023 Junior P4

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

1

NMTC 2023 Junior P5

a,b,c

are positive reals satisfying

\frac{2}{5} \leq c \leq \min{a,b}

ac \geq \frac{4}{15}

bc \geq \frac{1}{5}

Find the maximum value of

\left(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\right)

1

NMTC 2023 FINALS P3 JUNIOR

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

i \in {1,2,3,...,8}

. Find the possible values of

a_1

1

NMTC JUNIOR 2023 FINALS P2

PQR

is an acute scalene triangle. The altitude

PL

and the bisector

RK

\angle QRP

H

L

QR

K

PQ

KM

is the altitude of triangle

PKR

PL

N

. The circumcircle of

\triangle NKR

QR

S

Q

SHK

is an isosceles triangle.

1

NMTC JUNIOR 2023 FINALS P1

m,n

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