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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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KJMO 2001 P1
RL_parkgong_0106   1
N 7 minutes ago by JH_K2IMO
Source: KJMO 2001
A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is $141$. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.
1 reply
RL_parkgong_0106
Jun 29, 2024
JH_K2IMO
7 minutes ago
J
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Cotangential circels
CountingSimplex   5
N 10 minutes ago by rong2020
Let $ABC$ be a triangle with circumcenter $O$ and let the angle bisector of $\angle{BAC}$ intersect $BC$
at $D$. The point $M$ is such that $\angle{MCB}=90^o$ and $\angle{MAD}=90^o$. Lines $BM$ and $OA$ intersect at
the point $P$. Show that the circle centered at $P$ and passing through $A$ is tangent to segment
$BC$.
5 replies
CountingSimplex
Jun 23, 2020
rong2020
10 minutes ago
J
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Inspired by old results
sqing   3
N 17 minutes ago by sqing
Source: Own
Let $a,b,c $ be reals such that $a^2+b^2+c^2=3$ .Prove that
$$(1-a)(k-b)(1-c)+abc\ge -k$$Where $ k\geq 1.$
$$(1-a)(1-b)(1-c)+abc\ge -1$$$$(1-a)(1-b)(1-c)-abc\ge -\frac{1}{2}-\sqrt 2$$
3 replies
sqing
Today at 7:36 AM
sqing
17 minutes ago
J
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Inspired by Philippine 2025
sqing   1
N 22 minutes ago by sqing
Source: Own
Let $ a,b,c,d $ be real numbers . Prove that
$$\frac{(a-1)(b-3)(c-3)(d-1)}{ (a^2+3)(b^2+3)(c^2+3)(d^2+3)} \ge -\frac{7+4\sqrt 3}{144}$$$$\frac{(a-1)(b-2)(c-2)(d-1)}{ (a^2+3)(b^2+3)(c^2+3)(d^2+3)} \ge -\frac{11+4\sqrt 7}{432}$$


1 reply
sqing
37 minutes ago
sqing
22 minutes ago
J
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Combinatorics
P162008   0
an hour ago
$4$ girls and $4$ boys are to be seated in a line. Find the total number of ways such that boys and girls are alternate and a particular boy and girl are never adjacent to each other in any arrangement.
0 replies
P162008
an hour ago
0 replies
J
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Combinatorics
P162008   0
an hour ago
A cricket team comprising of $11$ players named $A,B,C,\cdots,J,K$ is to be sent for batting. If $A$ wants to bat before $J$ and $J$ wants to bat after $G.$ Then find the total number of batting orders if other players could go in any order.
0 replies
P162008
an hour ago
0 replies
J
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[PMO25 Qualifying II.8] A Square Can't Be A Floor
kae_3   2
N 2 hours ago by tapilyoca
Determine the largest perfect square less than $1000$ that cannot be expressed as $\lfloor x\rfloor + \lfloor 2x\rfloor + \lfloor 3x\rfloor + \lfloor 6x\rfloor$ for some positive real number $x$.

Answer Confirmation
2 replies
kae_3
Feb 21, 2025
tapilyoca
2 hours ago
J
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solve in R
zolfmark   1
N 2 hours ago by Mathzeus1024
x+1/y=9/2 and y+1/z=11/4 and z+1/x=12/5
1 reply
zolfmark
Feb 16, 2016
Mathzeus1024
2 hours ago
J
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not so hard problem (own)
BinariouslyRandom   1
N 3 hours ago by BinariouslyRandom
An open rectangular box is made from exactly $900 m^2$ of cardboard, with 5 rectangular faces instead of 6 and there is no leftover cardboard. What is the largest volume this box can hold given that $100 m^2$ of cardboard is needed to cover the box?
1 reply
BinariouslyRandom
3 hours ago
BinariouslyRandom
3 hours ago
J
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Minimize z.
Entrepreneur   3
N 3 hours ago by no_room_for_error
Minimize $z = 6x + 3y.$ Subject to the constraints:
$$\begin{cases} 4x+y\ge80,\\ x+5y \ge115,\\ 3x+2y\le150,\\ x,y\ge0. \end{cases}$$
3 replies
Entrepreneur
May 21, 2025
no_room_for_error
3 hours ago
J
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Inequalities
sqing   18
N Today at 7:48 AM by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
18 replies
sqing
May 21, 2025
sqing
Today at 7:48 AM
J
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Inequalities
sqing   18
N Today at 7:20 AM by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
18 replies
sqing
May 13, 2025
sqing
Today at 7:20 AM
J
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helpppppppp me
stupid_boiii   1
N Today at 6:56 AM by vanstraelen
Given triangle ABC. The tangent at ? to the circumcircle(ABC) intersects line BC at point T. Points D,E satisfy AD=BD, AE=CE, and ∠CBD=∠BCE<90 ∘ . Prove that D,E,T are collinear.
1 reply
stupid_boiii
Yesterday at 4:22 AM
vanstraelen
Today at 6:56 AM
J
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Algebra Polynomials
Foxellar   2
N Today at 5:43 AM by Foxellar
The real root of the polynomial \( p(x) = 8x^3 - 3x^2 - 3x - 1 \) can be written in the form
\[ \frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}, \]where \( a, b, \) and \( c \) are positive integers. Find the value of \( a + b + c \).
2 replies
Foxellar
Today at 4:52 AM
Foxellar
Today at 5:43 AM
J
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J
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Volume of right parallelepiped with minimal surface area
WakeUp   2
N Feb 2, 2011 by m.candales
Source: Romanian JBMO TST 2001
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
2 replies
WakeUp
Jan 19, 2011
m.candales
Feb 2, 2011
J
Volume of right parallelepiped with minimal surface area
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Reveal topic tags
geometry
number theory proposed
number theory
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G H BBookmark kLocked kLocked NReply
Source: Romanian JBMO TST 2001
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WakeUp
1347 posts
WakeUp
#1 h  Jan 19, 2011, 5:53 PM • 2 Y
Y by Adventure10, Mango247
Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
Z K Y
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mszew
1046 posts
mszew
#2 h  Jan 28, 2011, 7:26 PM • 2 Y
Y by Adventure10, Mango247
Volume: $V=abc\ge 1001$
half Area $A_2=\frac{A}{2}=ab+bc+ca$

WLOG: $a \le b \le c$

$a=b=10$ $c=11$, $V=1100$, $A_2=320$ Then $a \le 10$

If $a\le 3$ then $334 \le bc \le A_2$
then $a \in \{4,5,6,7,8,9,10\}$

Notice that $\frac{1001}{a} \le bc$
and $A_2=a(b+c)+bc$
Checking some cases to find $(a,b,c)=(7,12,12)$
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m.candales
186 posts
m.candales
#3 h  Feb 2, 2011, 4:12 PM • 2 Y
Y by Adventure10, Mango247
The right answer is $(8,9,14)$
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