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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
Yesterday at 11:16 PM
0 replies
J
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Maximum value of n
Ecrin_eren   0
33 minutes ago


"Let M be the set {1, 2, 3, ..., 2025}. Jack selects n different subsets of M. If the union of any two subsets Jack selects is never equal to M, what is the maximum possible value of n?"

0 replies
Ecrin_eren
33 minutes ago
0 replies
J
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How many integer pairs
Ecrin_eren   0
38 minutes ago

"Let m and n be integers. How many different integer pairs (m, n) satisfy the equation m^3 - 3m^2n + 4n^3 = 44?"

0 replies
Ecrin_eren
38 minutes ago
0 replies
J
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How many triangles
Ecrin_eren   0
an hour ago


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


0 replies
Ecrin_eren
an hour ago
0 replies
J
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Find the functions
Ecrin_eren   2
N 2 hours ago by Ecrin_eren
"Find all differentiable functions f that satisfy the condition f(x) + f(y) = f((x + y) / (1 - xy)) for all x, y ∈ R, where xy ≠ 1."
2 replies
Ecrin_eren
Yesterday at 8:58 PM
Ecrin_eren
2 hours ago
J
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All possible values of k
Ecrin_eren   1
N 2 hours ago by Ecrin_eren


The roots of the polynomial
x³ - 2x² - 11x + k
are r₁, r₂, and r₃.

Given that
r₁ + 2r₂ + 3r₃ = 0,
what is the product of all possible values of k?

1 reply
Ecrin_eren
4 hours ago
Ecrin_eren
2 hours ago
J
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Angle AEB
Ecrin_eren   1
N 2 hours ago by Ecrin_eren
In triangle ABC, the lengths |AB|, |BC|, and |CA| are proportional to 4, 5, and 6, respectively. Points D and E lie on segment [BC] such that the angles ∠BAD, ∠DAE, and ∠EAC are all equal. What is the measure of angle ∠AEB in degrees?

1 reply
Ecrin_eren
3 hours ago
Ecrin_eren
2 hours ago
J
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20 fair coins are flipped, N of them land heads 2024 TMC AIME Mock #6
parmenides51   6
N 3 hours ago by MelonGirl
$20$ fair coins are flipped. If $N$ of them land heads, find the expected value of $N^2$.
6 replies
parmenides51
Apr 26, 2025
MelonGirl
3 hours ago
J
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China MO 1996 p1
math_gold_medalist28   0
3 hours ago
Let ABC be a triangle with orthocentre H. The tangent lines from A to the circle with diameter BC touch this circle at P and Q. Prove that H, P and Q are collinear.
0 replies
math_gold_medalist28
3 hours ago
0 replies
J
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A problem with a rectangle
Raul_S_Baz   14
N 4 hours ago by george_54
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
14 replies
Raul_S_Baz
Apr 26, 2025
george_54
4 hours ago
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Inequalities
sqing   16
N 4 hours ago by sqing
Let $ a,b>0 $ and $ a+ b^2=\frac{3}{4} $.Prove that
$$ \frac{1}{a^3(a+b)} + \frac{2}{b^3(2b+1)} + \frac{16}{2a+1} \geq 24$$Let $ a,b>0 $ and $a^2+b^2=\frac{1}{2} $.Prove that
$$ \frac{1}{a^3(a+b)} + \frac{2}{b^3(2b+1)} + \frac{16}{2a+1} \geq 24$$
16 replies
sqing
Nov 29, 2024
sqing
4 hours ago
J
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Sum of solutions
Ecrin_eren   1
N 4 hours ago by Mathzeus1024

"[(x - 2)^2 + 4] * (x + (1/x)) = 10. What is the sum of the elements in the solution set of this equation?

1 reply
Ecrin_eren
5 hours ago
Mathzeus1024
4 hours ago
J
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Value of expression
Ecrin_eren   0
5 hours ago
Let a be a root of the equation x^3-x-1=0 , with a>1
What is the value of the expression:
∛(3a^2 - 4a) + ∛(3a^2 + 4a + 2)?
0 replies
Ecrin_eren
5 hours ago
0 replies
J
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Inequalities
sqing   5
N Today at 4:55 AM by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$https://artofproblemsolving.com/community/c6h107534p608181:
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 1+\frac{a+b}{b+c}+\frac{b+c}{a+b}$$
5 replies
sqing
Yesterday at 12:20 AM
sqing
Today at 4:55 AM
J
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Inequality
Ecrin_eren   1
N Today at 1:17 AM by sqing


Let a, b, c be positive real numbers. Prove the inequality:

sqrt(a² - ab + b²) + sqrt(b² - bc + c²) ≥ sqrt(a² + ac + c²)



1 reply
Ecrin_eren
Yesterday at 8:47 PM
sqing
Today at 1:17 AM
J
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J
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Combinatoric
spiderman0   3
N Apr 27, 2025 by MathBot101101
Let $ S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$
3 replies
spiderman0
Apr 22, 2025
MathBot101101
Apr 27, 2025
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spiderman0
11 posts
spiderman0
#1 h  Apr 22, 2025, 7:46 AM
Y by
Let $ S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$
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MathBot101101
17 posts
MathBot101101
#2 h  Apr 23, 2025, 6:44 AM
Y by
We want good sets from a subset T with n elements satisfying that equation.

Solved the inequality to get (just square both sides cuz they're positive, ig)
\frac{9a}{25} < b < a

Now we want the maximum number of elements any bad set can have. Suppose a bad set
P={x_1, x_2, ..., x_{m}} and x_{i}>x_{i-1} for all i belonging to {1, 2, ..., m}
So, x_{i+1} >= \frac{25}{9} x_{i}

x_1=1
x_2= ceiling of (\frac{25}{9}*1)= 3
and so on till we get an x_{k} > 2024

k comes out to be 8.

Therefore your answer is 9. : )

(PS: PLEASEE Latex-ify this, i can't)
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persamaankuadrat
155 posts
persamaankuadrat
#3 h  Apr 26, 2025, 12:29 AM
Y by
How did you derive $\frac{9a}{25} < b$ ?
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MathBot101101
17 posts
MathBot101101
#4 h  Apr 27, 2025, 3:38 AM
Y by
square both sides and then solve and then take cases, ig
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