MathDB

1

Easy Geometry

\triangle ABC

which

\angle C=90

, and

\bar{AB}=36

On the circumcircle of

\triangle ABC

, there is \overarc{BC} which does not include point

A

. D is on \overarc{BC}. It satisfies

2\times\angle CAD = \angle BAD

E: \bar{AD}\cap\bar{BC}

\bar{AE}=20

Find

\bar{BD}^2

19.

1

algebra finding maximum

For all integers

{a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}

, find the maximum of

{a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95}

\bigstar

1)

{a}_{0}={a}_{100}=0

2) for all

i=0,1,\cdot \cdot \cdot 99,

|{a}_{i+1}-{a}_{i}|\le1

3)

{a}_{10}={a}_{90}

18.

1

Combinatorics

As shown in the following figure, there is a line segment consisting of five line segments

AB, BC, CD, DE, and EA

and

10

intersection points of these five line segments. Find the number of ways to write

1

or

2

at each of the

10

vertices so that the following conditions are satisfied.

\bigstar

The sum of the four numbers written on each line segment

AB, BC, CD, DE, and EA

is the same.

17.

1

Number Theory with modulars

Find the number of

n

that follow the following:

\bigstar

The number of integers

(x,y,z)

following this equation is not a multiple of 4.

2n=x^2+2y^2+2x^2+2xy+2yz

16.

1

Geometry

There is an Equilateral trapezoid

ABCD.

\bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108.

M

is the middle point of

\bar {AB}

, and point

P

on

\bar{AM}

follows that

\bar {AP}

=10. The foot of perpendicular dropped from

P

to

\bar {BD}

is

E

.

\bar{AC} \cap \bar{BD}

is

F

. Point

X

is on

\bar {AF}

which follows

\bar{MX}=\bar{ME}

Find

\bar{AX} \times \bar{AF}

15.

1

combinatorics counting

In the following illustration, starting from point X, we move one square along the segment until we arrive at point Y. Calculate the number of times a point has passed once and does not pass again, from X to Y. (However, starting point X is considered to have passed.)

14.

1

Algebra with numbers

Find the number of positive integer

x

that has

{a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20}

which follows the following (

x \ge 1000

)
1)

{a}_{1}=2, {a}_{2}=1, {a}_{3}=x

2) for positive integer

n

, (

4 \le n \le 20

),

{a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}}

13.

1

number theory and gauss

Find the number of positive integer n, which follows the following

\bigstar

n=[\frac{m^3}{2024}]

n

has a positive integer

m

that follows this equation (

m \le 1000

)

12.

1

Algebra Finding Maximum

For reals

x,y

, find the maximum of A.

A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12}

11.

1

Geometry finding area

There is a square

ABCD.

P

is on

\bar{AB}

, and

Q

is on

\bar{AD}

They follow

\bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5}

Let

H

be the foot of the perpendicular point from

A

to

\bar{PD}

If

|\triangle APH|=20

, Find the area of

\triangle HCQ

.

10.

1

Combinatorics counting

Find the number of cases in which one of the numbers 1, 2, 3, 4, and 5 is written at each vertex of an equilateral triangle so that the following conditions are satisfied. (However, the same number is counted as one when rotated, and the same number can be written multiple times.)

\bigstar

The product of the two numbers written at each end of the sides of an equilateral triangle is an even number.

9.

1

numbers of divisors

Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

8.

1

combinatorics counting

Find the number of 4 digit positive integers '

n

' that follow these.
1) the number of digit

\le

6 2)

3 \mid n

, but

6 \nmid n

7.

1

Geometry, finding area.

There are four collinear spots:

A,B,C,D

\bar{AB}=\bar{BC}=\frac{\bar{CD}}{4}=\sqrt{5}

There are two circles; One which has

\bar{AC}

as a diameter, and the other having

\bar{BD}

as a diameter. Let's put

\odot (AC) \cap \odot (BD) = E,F

Let's put the area of

EAFD

S

Find

S^2

.

6.

1

algebra with gaus

Find the number of

x

which follows the following :

x-\frac{1}{x}=[x]-[\frac{1}{x}]

( \frac{1}{100} \le x \le {100} )

5.

1

Number Theory

Find the addition of all positive integers n that follows the following:

\frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}}

is an integer.

4.

1

Combinatorics

There is a shape like this (Attachment down below)
Find the number of triangles made by choosing 3 vertex from the 8 vertex in the attachment.

3.

1

Number Theory + Combinatorics

Find the number of positive integers (m,n) which follows the following:
1) m

2

1

Geometry with calculation

There is an isosceles triangle which follows the following:

\bar{AB}=\bar{AC}=5, \bar{BC}=6

D,E are points on

\bar{AC}

which follows

\bar{AD}=1, \bar{EC}=2

If the extent of

\triangle

BDE = S, Find 15S.

1.

1

Easy Fraction Calculating

Find this:

(1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024})

2024 Korea Junior Math Olympiad (First Round)

Subcontests

Easy Geometry

algebra finding maximum

Combinatorics

Number Theory with modulars

Geometry

combinatorics counting

Algebra with numbers

number theory and gauss

Algebra Finding Maximum

Geometry finding area

Combinatorics counting

numbers of divisors

combinatorics counting

Geometry, finding area.

algebra with gaus

Number Theory

Combinatorics

Number Theory + Combinatorics

Geometry with calculation

Easy Fraction Calculating