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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Indonesia Regional MO 2019 Part A
parmenides51   15
N 11 minutes 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 ....
15 replies
parmenides51
Nov 11, 2021
Rohit-2006
11 minutes ago
J
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A Complex Product
Saucepan_man02   10
N an hour ago by alexheinis
Calculate $\prod_{n \ge j>i \ge 1} (\epsilon^j-\epsilon^i)$ where $\epsilon = e^{\frac{2 \pi}{n}i}$.
10 replies
Saucepan_man02
Today at 2:15 AM
alexheinis
an hour ago
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Inequalities
lgx57   3
N 3 hours ago by lgx57
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
3 replies
lgx57
Mar 19, 2025
lgx57
3 hours ago
J
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Inequalities
sqing   0
5 hours ago
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
0 replies
sqing
5 hours ago
0 replies
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functional equation in R2
jasperE3   2
N Mar 31, 2025 by alexheinis
Find all functions $f:\mathbb R\times\mathbb R\to\mathbb R$ such that:
$a)\enspace f(x+z,y+z)=f(x,y)+z$
$b)\enspace f(xw,yw)=wf(x,y)$
both hold $\forall w,x,y,z\in\mathbb R,w\ne0$.
2 replies
jasperE3
Mar 27, 2021
alexheinis
Mar 31, 2025
J
functional equation in R2
G H J
Reveal topic tags
algebra
functional equation
function
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jasperE3
11214 posts
jasperE3
#1 h  Mar 27, 2021, 10:18 PM
Y by
Find all functions $f:\mathbb R\times\mathbb R\to\mathbb R$ such that:
$a)\enspace f(x+z,y+z)=f(x,y)+z$
$b)\enspace f(xw,yw)=wf(x,y)$
both hold $\forall w,x,y,z\in\mathbb R,w\ne0$.
This post has been edited 1 time. Last edited by jasperE3, Mar 27, 2021, 10:20 PM
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jasperE3
11214 posts
jasperE3
#2 h  Mar 31, 2025, 6:58 AM
Y by
Bump (this problem is easy enough)
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alexheinis
10544 posts
alexheinis
#3 h  Mar 31, 2025, 9:14 AM
Y by
Taking $x=y=0,w=2$ we find $f(0,0)=0$. Define $g(x):=f(x,y)$ then with a) we find $f(p,q)=q+g(p-q)$.
Substituting this in b) we find $g((x-y)w)=wg(x-y)$ when $w\not=0$. Also true for $w=0$ hence $g(tw)=wg(t)$ for all $t,w$.
With $t=1$ we find $g(w)=cw$. Finally we find $f(x,y)=y+c(x-y)$ and these linear maps work.
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