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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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
May 1, 2025
0 replies
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exist solutions?
teomihai   6
N an hour ago by iwastedmyusername
Find how many perfect squares of five different digits there are, with elements from the set ${0,1,4,6,9}$.
6 replies
teomihai
Yesterday at 5:04 PM
iwastedmyusername
an hour ago
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A pentagon inscribed in a circle of radius √2
tom-nowy   6
N an hour ago by anticodon
Can a pentagon with all rational side lengths be inscribed in a circle of radius $\sqrt{2}$ ?
6 replies
tom-nowy
May 6, 2025
anticodon
an hour ago
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Menelau's theorem
noneofyou34   6
N 2 hours ago by Shan3t
Please can someone help me prove that orthocenter of a triangle exists by using Menelau's Theorem!
6 replies
noneofyou34
Yesterday at 5:52 PM
Shan3t
2 hours ago
J
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a,b,c irrational, f(x)=ax^2+bx+c : [-1,1] to [-1,1] surjective
tom-nowy   0
2 hours ago
Consider a quadratic function $f(x) = ax^2 + bx + c$, where the coefficients $a, b,$ and $c$ are all irrational numbers.
Is it possible for this function to have a maximum value of $1$ and a minimum value of $-1$ over the interval $[-1, 1]$?
0 replies
tom-nowy
2 hours ago
0 replies
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Math analytical geometry
JDog22   2
N 5 hours ago by Alex-131
Could you please tell me if this is correct:
If you take the position vector of an arbitrary point P, which is known to lie on the plane E, then the dot product of P and the normal vector n always results in the same number, regardless of the coordinates of point P. As long as the point lies on the plane, this calculation always gives the same number. Is that true?
2 replies
JDog22
5 hours ago
Alex-131
5 hours ago
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Hard Inequality
William_Mai   14
N Yesterday at 5:34 PM by hi2024IMOp14
Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$.
Find the minimum value of $P = ab + 2bc + 3ca$.

Source: Pham Le Van
14 replies
William_Mai
May 3, 2025
hi2024IMOp14
Yesterday at 5:34 PM
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Geometry
MTA_2024   7
N Yesterday at 4:56 PM by hi2024IMOp14
Let $ABC$ be a triangle such that $AB=3$,$BC=5$ and $AC=6$.Let $D$ be a point on side $AC$ and $E$ one on side $BC$ so that the line $DE$ is tangent to the incircle of $\triangle ABC$ .
Evaluate the perimeter of triangle $\triangle CDE$.
7 replies
MTA_2024
Wednesday at 6:52 PM
hi2024IMOp14
Yesterday at 4:56 PM
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four point lie on circle
Kizaruno   0
Yesterday at 3:29 PM
Let triangle ABC be inscribed in a circle with center O. A line d intersects sides AB and AC at points E and D, respectively. Let M, N, and P be the midpoints of segments BD, CE, and DE, respectively. Let Q be the foot of the perpendicular from O to line DE. Prove that the points M, N, P, and Q lie on a circle.
0 replies
Kizaruno
Yesterday at 3:29 PM
0 replies
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Inequalities
sqing   0
Yesterday at 2:42 PM
Let $ a,b>0, a^2+ab+b^2 \geq 6 $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$a^4+ab+b^4 \leq 10$$Let $ a,b>0, a^2+ab+b^2 \geq \frac{15}{2} $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$-\frac{1}{8}\leq a^4-ab+b^4\leq 10$$
0 replies
sqing
Yesterday at 2:42 PM
0 replies
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Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)
ThisIsJoe   0
Yesterday at 2:02 PM
For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?
0 replies
ThisIsJoe
Yesterday at 2:02 PM
0 replies
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Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   1
N Yesterday at 1:10 PM by pooh123
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$Proposed by Ngo Van Trang, Vietnam
1 reply
Adventure1000
Wednesday at 4:10 PM
pooh123
Yesterday at 1:10 PM
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one nice!
MihaiT   3
N Yesterday at 12:53 PM by Pin123
Find positiv integer numbers $(a,b) $ s.t. $\frac{a}{b-2} $ and $\frac{3b-6}{a-3}$ be positiv integer numbers.
3 replies
MihaiT
Jan 14, 2025
Pin123
Yesterday at 12:53 PM
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Acute Angle Altitudes... say that ten times fast
Math-lover1   1
N Yesterday at 12:29 PM by pooh123
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$, respectively. Suppose that $AC-AB=36$ and $DC-DB=24$. Compute $EC-EB$.
1 reply
Math-lover1
Wednesday at 11:30 PM
pooh123
Yesterday at 12:29 PM
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Find a and b such that a^2 = (a-b)^3 + b and a and b are coprimes
picysm   2
N Yesterday at 8:28 AM by picysm
it is given that a and b are coprime to each other and a, b belong to N*
2 replies
picysm
Apr 25, 2025
picysm
Yesterday at 8:28 AM
J
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J
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Binomial Sum
P162008   2
N Apr 27, 2025 by IbrahimNadeem
The numbers $p$ and $q$ are defined in the following manner:

$p = 99^{98} - \frac{99}{1} 98^{98} + \frac{99.98}{1.2} 97^{98} - \frac{99.98.97}{1.2.3} 96^{98} + .... + 99$

$q = 99^{100} - \frac{99}{1} 98^{100} + \frac{99.98}{1.2} 97^{100} - \frac{99.98.97}{1.2.3} 96^{100} + .... + 99$

If $p + q = k(99!)$ then find the value of $\frac{k}{10}.$
2 replies
P162008
Apr 23, 2025
IbrahimNadeem
Apr 27, 2025
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Binomial Sum
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P162008
186 posts
P162008
#1 h  Apr 23, 2025, 12:03 PM • 1 Y
Y by girishpimoli
The numbers $p$ and $q$ are defined in the following manner:

$p = 99^{98} - \frac{99}{1} 98^{98} + \frac{99.98}{1.2} 97^{98} - \frac{99.98.97}{1.2.3} 96^{98} + .... + 99$

$q = 99^{100} - \frac{99}{1} 98^{100} + \frac{99.98}{1.2} 97^{100} - \frac{99.98.97}{1.2.3} 96^{100} + .... + 99$

If $p + q = k(99!)$ then find the value of $\frac{k}{10}.$
This post has been edited 1 time. Last edited by P162008, Apr 23, 2025, 12:03 PM
Reason: Typo
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vanstraelen
9013 posts
vanstraelen
#2 h  Apr 27, 2025, 8:30 AM • 1 Y
Y by P162008
$p=\sum_{k=0}^{98}(-1)^{k}\left(\stackrel{99}{k}\right)(99-k)^{98}=0$.

$q=\sum_{k=0}^{98}(-1)^{k}\left(\stackrel{99}{k}\right)(99-k)^{100}=50 \cdot 99 \cdot 99!$
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IbrahimNadeem
888 posts
IbrahimNadeem
#3 h  Apr 27, 2025, 8:54 AM
Y by
vanstraelen wrote:
$p=\sum_{k=0}^{98}(-1)^{k}\left(\stackrel{99}{k}\right)(99-k)^{98}=0$.

$q=\sum_{k=0}^{98}(-1)^{k}\left(\stackrel{99}{k}\right)(99-k)^{100}=50 \cdot 99 \cdot 99!$

can someone please explain this for me?
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