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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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Ihave a minor issue.
CovertQED   1
N 2 hours ago by vanstraelen
The area of triangle ABC is 18，sin2A +sin2B =4sinAsinB.Find the minimum perimeter of triangle ABC.
1 reply
CovertQED
Today at 1:52 PM
vanstraelen
2 hours ago
J
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geometry parabola problem
smalkaram_3549   8
N 3 hours ago by jb2015007
How would you solve this without using calculus?
8 replies
smalkaram_3549
Yesterday at 9:52 PM
jb2015007
3 hours ago
J
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Indonesia Regional MO 2019 Part A
parmenides51   13
N Today at 3:09 PM 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 ....
13 replies
parmenides51
Nov 11, 2021
SomeonecoolLovesMaths
Today at 3:09 PM
J
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Diophantine Equation in (x^2+4) set
Johann Peter Dirichlet   0
Today at 3:01 PM
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
Today at 3:01 PM
0 replies
J
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angle of isosceles wanted, 4HK=AB, altitudes (2021 Euler Olympiad Remote 2.4)
parmenides51   1
N Feb 14, 2021 by natmath
$AH$ is the altitude of an isosceles triangle $ABC$ ($AB = BC$). $HK$ is the altitude of the triangle $AHB$. It turned out that $4HK = AB$. What could be the measure of the angle $ABC$?
1 reply
parmenides51
Feb 14, 2021
natmath
Feb 14, 2021
J
angle of isosceles wanted, 4HK=AB, altitudes (2021 Euler Olympiad Remote 2.4)
G H J
Reveal topic tags
geometry
isosceles
equal segments
angles
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parmenides51
30630 posts
parmenides51
#1 h  Feb 14, 2021, 10:38 AM
Y by
$AH$ is the altitude of an isosceles triangle $ABC$ ($AB = BC$). $HK$ is the altitude of the triangle $AHB$. It turned out that $4HK = AB$. What could be the measure of the angle $ABC$?
This post has been edited 1 time. Last edited by parmenides51, Feb 14, 2021, 10:38 AM
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natmath
8219 posts
natmath
#2 h  Feb 14, 2021, 6:46 PM • 2 Y
Y by Mango247, Mango247
$\Delta AHB$ is a right triangle with right angle at $H$.
If $AK=x$ and $BK=y$, we have
$$x+y=4\sqrt{xy}$$$$x^2+y^2+2xy=16xy$$$$x^2+y^2-14xy=0$$Define $u=\frac{x}{y}$
$$u^2-14u+1=0$$$$u=7\pm 4\sqrt{3}$$Note that $\tan\angle B=\sqrt{u}$, which is
$$\sqrt{7\pm 4\sqrt{3}}$$$$2\pm\sqrt{3}$$
This means $\angle B=15,75$
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