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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Indonesia Regional MO 2019 Part A
parmenides51   12
N a few seconds ago by SomeonecoolLovesMaths
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 ....
12 replies
+1 w
parmenides51
Nov 11, 2021
SomeonecoolLovesMaths
a few seconds ago
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high school math
aothatday   5
N 2 minutes ago by oz.the.wizard
Let $x_n$ be a positive root of the equation $x_n^n=x^2+x+1$. Prove that the following sequence converges: $n^2(x_n-x_{ n+1})$
5 replies
aothatday
Apr 10, 2025
oz.the.wizard
2 minutes ago
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Diophantine Equation in (x^2+4) set
Johann Peter Dirichlet   0
6 minutes ago
Let $S=\{n^2+4 | n \in \mathbb{Z}\}$.

Find all $p,q,r \in S$ so that $pq-r=4$.
0 replies
Johann Peter Dirichlet
6 minutes ago
0 replies
J
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easy problem
lgx57   1
N an hour ago by rchokler
Let $x+y=3$ , $\frac{1}{x^2+y}+\frac{1}{x+y^2}=\frac{1}{2}$. Find the value of $x^5+y^5$.
1 reply
lgx57
an hour ago
rchokler
an hour ago
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Ihave a minor issue.
CovertQED   0
an hour ago
The area of triangle ABC is 18，sin2A +sin2B =4sinAsinB.Find the minimum perimeter of triangle ABC.
0 replies
CovertQED
an hour ago
0 replies
J
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one very nice!
MihaiT   1
N 2 hours ago by MihaiT
Given $m_1$ weights, each weighing $k_1$ and another $m_2$ weights with $k_2$ each. Write a algorithm that determines the ways in which a scale can be balanced with a weight $X$ on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers $m_1,k_1,m_2,k_2,X$. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?

One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
1 reply
MihaiT
Mar 31, 2025
MihaiT
2 hours ago
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Geometry problem about Euler line
lgx57   2
N 3 hours ago by pooh123
If the Euler line of a triangle is parallel to one side of the triangle, what is the relationship between the sides of this triangle?

The relationship between the angles of this triangle
2 replies
lgx57
Apr 9, 2025
pooh123
3 hours ago
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Inequalities
sqing   5
N 6 hours ago by sqing
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ - \frac{1681}{3}\leq ab - cd \leq 820$$$$ - \frac{16564}{9}\leq ac -bd \leq 420$$$$ - \frac{10201}{48}\leq ad- bc \leq\frac{1681}{3}$$
5 replies
sqing
Yesterday at 3:53 AM
sqing
6 hours ago
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JEE Related ig?
mikkymini2   10
N Today at 4:08 AM by Idiot_of_the64squares
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
10 replies
mikkymini2
Apr 10, 2025
Idiot_of_the64squares
Today at 4:08 AM
J
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Inequalities
sqing   0
Today at 3:33 AM
Let $ a,b,c\in [0,1] $ . Prove that
$$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$$$$(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$$
0 replies
sqing
Today at 3:33 AM
0 replies
J
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lcm(1,2,3,...,n)
lgx57   5
N Today at 3:09 AM by Kempu33334
Let $M=\operatorname{lcm}(1,2,3,\cdots,n)$.Estimate the range of $M$.
5 replies
lgx57
Apr 9, 2025
Kempu33334
Today at 3:09 AM
J
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Inequality
math2000   7
N Today at 2:59 AM by imnotgoodatmathsorry
Let $a,b,c>0$.Prove that $\dfrac{1}{(a+b)\sqrt{(a+2c)(b+2c)}}>\dfrac{3}{2(a+b+c)^2}$
7 replies
math2000
Jan 22, 2021
imnotgoodatmathsorry
Today at 2:59 AM
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How to prove one-one function
Vulch   5
N Today at 2:56 AM by jasperE3
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!
5 replies
Vulch
Yesterday at 8:03 PM
jasperE3
Today at 2:56 AM
J
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Let a,b,c > 0 such that a+b+c=3. Prove that $ \frac{a^2}{a^2-2a+4} + \frac{b^2
bo_ngu_toan   3
N Today at 2:13 AM by imnotgoodatmathsorry
Let a,b,c > 0 such that a+b+c=3. Prove that $ \frac{a^2}{a^2-2a+4} + \frac{b^2}{b^2-2b+4} + \frac{c^2}{c^2-2c+4} \leq 1$
3 replies
bo_ngu_toan
Jun 4, 2023
imnotgoodatmathsorry
Today at 2:13 AM
J
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J
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AB/AD wanted, BC=2AC, BD=3DC (2018 Moldova NMO 7.2)
parmenides51   7
N Mar 25, 2021 by SatisfiedMagma
Consider the triangle $ABC$ with $BC=2AC$. Let $D \in (BC)$ such that $BD=3DC$. Find the value of the ratio $\frac{AB}{AD}$.
7 replies
parmenides51
Mar 17, 2021
SatisfiedMagma
Mar 25, 2021
J
AB/AD wanted, BC=2AC, BD=3DC (2018 Moldova NMO 7.2)
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Reveal topic tags
geometry
ratio
equal segments
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parmenides51
30630 posts
parmenides51
#1 h  Mar 17, 2021, 2:45 PM
Y by
Consider the triangle $ABC$ with $BC=2AC$. Let $D \in (BC)$ such that $BD=3DC$. Find the value of the ratio $\frac{AB}{AD}$.
This post has been edited 1 time. Last edited by parmenides51, Mar 17, 2021, 2:45 PM
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SatisfiedMagma
457 posts
SatisfiedMagma
#2 h  Mar 17, 2021, 3:40 PM
Y by
This is a wrong problem, pretty sure. You can't determine the ratio I guess... Waiting for suggestions
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vanstraelen
8956 posts
vanstraelen
#3 h  Mar 17, 2021, 4:08 PM • 1 Y
Y by A.L.E.X
$C(0,0),B(2b,0),A(b\cos \alpha,b\sin \alpha)$ and $D(\frac{b}{2},0)$.
Ratio $\frac{AB}{AD}=2$.
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MathArt4
3674 posts
MathArt4
#4 h  Mar 17, 2021, 4:12 PM
Y by
Whoops sniped!
Solution
This post has been edited 3 times. Last edited by MathArt4, Mar 25, 2021, 11:53 AM
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A.L.E.X
133 posts
A.L.E.X
#5 h  Mar 17, 2021, 4:15 PM
Y by
easy
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HamstPan38825
8857 posts
HamstPan38825
#6 h  Mar 17, 2021, 4:16 PM
Y by
@#2
Wait, why do you think it is wrong?

Personally I like setting the side lengths to be actual integers, then solving. :D
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franzliszt
23531 posts
franzliszt
#7 h  Mar 18, 2021, 12:35 AM
Y by
WLOG, let $BD=3,DC=1,AC=2,AB=x$. Now apply barycentric coordinates w.r.t. $\triangle ABC$. We have $A=(1,0,0),B=(0,1,0),C=(0,0,1),D=(0,1/4,3/4)$. Observe that $\overrightarrow{AD}=\vec{D}-\vec{A}=(-1,1/4,3/4)$. Then by the distance formula, $|AD|^2=-4^2\cdot\frac14\cdot\frac34-2^2\cdot-1\cdot\frac34-x^2\cdot-1\cdot\frac14 \iff AD=\frac{x}{2}$ and the answer is clearly $2$.
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SatisfiedMagma
457 posts
SatisfiedMagma
#8 h  Mar 25, 2021, 9:55 AM
Y by
[asy] draw((0,0)--(6,8)--(5,0)--cycle); draw((6,8)--(6,0)--cycle,dotted); draw((1.5,2)--(5,0)--cycle,dotted); draw((1.5,2)--(1.5,0)--cycle,dotted); draw((5,0)--(6,0)--cycle,dotted); label("$2x$",(2.7,4), W); label("$x$",(2.5,0), S); label("$z-x$",(5.5,0), S); label("$y$",(6,4), E); [/asy]
Just time pass...
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