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Solve an equation

lgx57 2

N 4 hours ago by lgx57

Find all positive integers $n$ and $x$ such that:
$2^{2n+1}-7=x^2$

2 replies

lgx57

Mar 12, 2025

lgx57

4 hours ago

Indonesia Regional MO 2019 Part A

parmenides51 17

N 4 hours ago by Rohit-2006

Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules

p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...

p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...

p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$ . The value of $(r-s)^2$ is ...

p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...

p5. A data with four real numbers $2n-4$ , $2n-6$ , $n^2-8$ , $3n^2-6$ has an average of $0$ and a median of $9/2$ . The largest number of such data is ...

p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$ . If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...

p7. Given a triangle $ABC$ , with $AB = 6$ , $AC = 8$ and $BC = 10$ . The points $D$ and $E$ lies on the line segment $BC$ . with $BD = 2$ and $CE = 4$ . The measure of the angle $\angle DAE$ is ...

p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$ . The value of $a_1a_2a_3...a_{2019}$ is ....

p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...

p10. Pairs of natural numbers $(m , n)$ which satisfies $m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$ are as many as ...

p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$ . Point $D$ lies on the side $BC$ so that $AB=CD$ . Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$ . The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$ . The large angle $\angle AEC$ is ...

p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$ . The largest value of $n$ is ....

p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....

p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....

p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....

17 replies

parmenides51

Nov 11, 2021

Rohit-2006

4 hours ago

How to prove one-one function

Vulch 6

N 4 hours ago by Vulch

Hello everyone,
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers $(1,2)$ from the domain $\Bbb R_{*},$ and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!

6 replies

Vulch

Apr 11, 2025

Vulch

4 hours ago

Inequalities

sqing 6

N 4 hours ago by sqing

Let $a,b,c> 0$ and $\frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$ . Prove that
$a+b+c\leq 3$

6 replies

sqing

Apr 4, 2025

sqing

4 hours ago

hard number theory

eric201291 0

4 hours ago

Prove:There are no integers x, y, that y^2+9998587980=x^3.

0 replies

eric201291

4 hours ago

0 replies

Amc 10 mock

Mathsboy100 3

N 5 hours ago by iwastedmyusername

let $\lfloor x \rfloor$ denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which $x^2 - 20\lfloor x \rfloor + 19 = 0$

3 replies

Mathsboy100

Oct 9, 2024

iwastedmyusername

5 hours ago

Inequalities

lgx57 4

N 5 hours ago by pooh123

Let $0 < a,b,c < 1$ . Prove that

$a(1-b)+b(1-c)+c(1-a)<1$

4 replies

lgx57

Mar 19, 2025

pooh123

5 hours ago

Let x,y,z be non-zero reals

Purple_Planet 3

N 5 hours ago by sqing

Let $x,y,z$ be non-zero real numbers. Define $E=\frac{|x+y|}{|x|+|y|}+\frac{|x+z|}{|x|+|z|}+\frac{|y+z|}{|y|+|z|}$ , then the number of all integers which lies in the range of $E$ is equal to.

3 replies

Purple_Planet

Jul 16, 2019

sqing

5 hours ago

Identity Proof

jjsunpu 2

N 6 hours ago by fruitmonster97

Hi this is my identity I name it Excalibur

I proved it already using induction what other ways?

2 replies

jjsunpu

Today at 10:35 AM

fruitmonster97

6 hours ago

Three 3-digit numbers

miiirz30 5

N 6 hours ago by fruitmonster97

Leonard wrote three 3-digit numbers on the board whose sum is $1000$ . All of the nine digits are different. Determine which digit does not appear on the board.

Proposed by Giorgi Arabidze, Georgia

5 replies

miiirz30

Mar 31, 2025

fruitmonster97

6 hours ago

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