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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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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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Geometry problem about Euler line
lgx57   2
N an hour 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
an hour ago
J
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Indonesia Regional MO 2019 Part A
parmenides51   10
N 3 hours ago by Mr.Awan
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 ....
10 replies
parmenides51
Nov 11, 2021
Mr.Awan
3 hours ago
J
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Inequalities
sqing   5
N 4 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
4 hours ago
J
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high school math
aothatday   3
N 4 hours ago by aothatday
Let $x_n$ be a positive root of the equation $x_n^n=x^2+x+1$. Prove that the following sequence converges: $n^2(x_n-x_{ n+1})$
3 replies
aothatday
Thursday at 2:20 PM
aothatday
4 hours ago
J
No more topics!
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Inequalites 1/4
nhathhuyyp5c   1
N Apr 1, 2025 by sqing
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=3$. Prove that $$a+ab+abc+abcd\leq 4$$
1 reply
nhathhuyyp5c
Apr 1, 2025
sqing
Apr 1, 2025
J
Inequalites 1/4
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inequalities
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nhathhuyyp5c
58 posts
nhathhuyyp5c
#1 h  Apr 1, 2025, 6:10 AM
Y by
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=3$. Prove that $$a+ab+abc+abcd\leq 4$$
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sqing
41548 posts
sqing
#3 h  Apr 1, 2025, 12:27 PM
Y by
nhathhuyyp5c wrote:
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=3$. Prove that $$a+ab+abc+abcd\leq 4$$
https://artofproblemsolving.com/community/c4h1061866p5234304
Let $a, b,c,d$ be non-negative numbers such that $4a+3b+2c+d=10.$ Prove that
$$a+ab+abc+abcd\leq 4.$$Let $a, b,c$ be non-negative numbers such that $3a+2b+c=6.$ Prove that
$$a+ab+abc\leq 3.$$
This post has been edited 1 time. Last edited by sqing, Apr 1, 2025, 12:33 PM
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