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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
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0 replies
jlacosta
Jun 2, 2025
0 replies
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Goodbye to this forum
RainbowSquirrel53B   22
N an hour ago by aops-fatima
:( I just realized that I'm going into high school and I'm too old to be here anymore. Goodbye to AMC 8, MathCOUNTS, BmMT, and all other competitions for only middle school.
22 replies
RainbowSquirrel53B
Jun 19, 2025
aops-fatima
an hour ago
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9 Favorite topic
A7456321   129
N 2 hours ago by M_a_t_h-M_i_n_i_o_n2012
What is your favorite math topic/subject?

If you don't know why you are here, go binge watch something!

If you forgot why you are here, go to a hospital! :)

If you know why you are here and have voted, maybe say why you picked the option that you picked in a response) :thumbup:

CLICK ON ME YOU KNOW YOU WANT TO

Timeline
129 replies
A7456321
May 23, 2025
M_a_t_h-M_i_n_i_o_n2012
2 hours ago
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AMC 8 info
VivaanKam   29
N 2 hours ago by M_a_t_h-M_i_n_i_o_n2012
Hi I will be attending the AMC 8 contest in 2026. How does it work? time? number of questions? points? scoring?
29 replies
VivaanKam
Jun 1, 2025
M_a_t_h-M_i_n_i_o_n2012
2 hours ago
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Arithmetic Grid Question
M_a_t_h-M_i_n_i_o_n2012   0
2 hours ago
In an arithmetic grid, adjacent numbers increase by a fixed integer \( a > 0 \) moving left to right within each row. Also, adjacent numbers increase by a fixed integer \( b > 0 \) moving top to bottom within each column. For example, the grid shown is a \( 3 \times 3 \) arithmetic grid with \( a = 2 \) and \( b = 5 \).

\[ \begin{array}{ccc} 1 & 3 & 5 \\ 6 & 8 & 10 \\ 11 & 13 & 15 \\ \end{array} \]
Suppose that an \( 8 \times 8 \) arithmetic grid has a 1 in the top left corner, and a number less than 75 in the bottom right corner. How many such grids have a 45 somewhere in column 5?
(A)6 (B)3 (C)7 (D)5 (E)4

So that's the problem, and as we can see, it's asking for the number of grids that include the number 45 somewhere in column 5. I solved this by substituting, so for example, the second box in the first row will be 1+a, the second box in the second row will be 1+a+b, and so on.
Then I just brought it to the fifth column, and I called the box in the fifth column in the first row 1+4a, and the box in the second row in the fifth column 1+4a+b, and so on.
After that I just used diophantine equations to solve and see if each box was capable of having a 45, and basically the only boxes that could have 45 is:
1+4a+5b(third last box)-total of 2 ways of happening
1+4a+6b(second last box)-total of 3 ways of happening
1+4a+7b(last box)-total of 1 way of happening
Then, 2+3+1=6, so my answer is 6.
However, as I was on the internet looking at how other people solved this problem, a lot of people chose 3, because there are three places in column 5 where the number 45 could be. I get what they're saying, but this problem is asking for the number of grids, and when the values are different in the grid, wouldn't they be considered different? So, I just wanted to ask, what's your answer for this problem?
(Also, for those who know, this problem is Problem 25 of 2025 Gauss 7, University of Waterloo, and the proper solutions are not up yet at the time of posting this)
0 replies
M_a_t_h-M_i_n_i_o_n2012
2 hours ago
0 replies
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[Own problem] Extended triangle cevians
aops-g5-gethsemanea2   1
N Jun 11, 2025 by aops-g5-gethsemanea2
Let $\triangle ABC$ be a triangle where $AB=AC=5$ and $BC=6$, and $\omega$ be the circumcircle of $\triangle ABC$. Then, let $D$ be on $\overline{AC}$ such that $\overline{BD}$ bisects $\angle ABC$ and $E$ be on $\overline{AB}$ such that $\overline{CE}\perp\overline{AB}$. Let $F$ be the intersection of $\overline{BD}$ and $\overline{CE}$, $G$ be the intersection of $\overrightarrow{BD}$ and $\omega$ that is not $B$, and $H$ be the intersection of $\overrightarrow{CE}$ and $\omega$ that is not $C$. Find $\frac{FE}{EH}$.
1 reply
aops-g5-gethsemanea2
Jun 11, 2025
aops-g5-gethsemanea2
Jun 11, 2025
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bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2
parmenides51   3
N May 17, 2025 by LeYohan
Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.
3 replies
parmenides51
Sep 20, 2021
LeYohan
May 17, 2025
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Acute Angle Altitudes... say that ten times fast
Math-lover1   1
N May 8, 2025 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
May 7, 2025
pooh123
May 8, 2025
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rhombus, line from bisector x circumcircle points (Puerto Rico TST 2024.7)
Equinox8   1
N Mar 15, 2025 by pooh123
Let $I$ be the incenter of $\triangle{ABC}$. Angle bisectors $BI$ and $CI$ intersect the circumcircle of $\triangle{ABC}$ at points $S$ and $T$, respectively ($S \neq B$ and $T \neq C$). Line $ST$ intersects sides $AB$ and $AC$ at points $K$ and $L$, respectively. Prove that quadrilateral $AKIL$ is a rhombus.

Note: A rhombus is a parallelogram with all sides equal.
1 reply
Equinox8
Mar 12, 2025
pooh123
Mar 15, 2025
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Geometry Angle Bisectors Problem
duttaditya18   3
N Mar 6, 2025 by S_14159
Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A, B$ and $C$ intersect $\Omega$ at $A_1, B_1$ and $C_1$, respectively, and the internal bisectors of angles $A_1, B_1$ and $C_1$ of the triangles $A_1 A_2 A_ 3$ intersect $\Omega$ at $A_2, B_2$ and $C_2$, respectively. If the smallest angle of the triangle $ABC$ is $40^{\circ}$, what is the magnitude of the smallest angle of the triangle $A_2 B_2 C_2$ in degrees?
3 replies
duttaditya18
Aug 11, 2019
S_14159
Mar 6, 2025
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[PMO24 Qualifying II.8] A Triangle's Package
kae_3   1
N Feb 18, 2025 by ehz2701
Let $ABC$ be a triangle such that the altitude from $A$, the median from $B$, and the internal angle bisector from $C$ meet at a single point. If $BC=10$ and $CA=15$, find $AB^2$.

Answer Confirmation
1 reply
kae_3
Feb 17, 2025
ehz2701
Feb 18, 2025
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Geometry
Osim_09   1
N Feb 8, 2025 by Osim_09
Let AB be a diameter of the circle Г and let C be a point on the circle, different from A and B.Denote by D the projection of C on AB and let w be a circle tangent to AD,CD and Г, touching Г at X.Prove that the angle bisectors of <AXB and <ACD meet on AB.
1 reply
Osim_09
Feb 3, 2025
Osim_09
Feb 8, 2025
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incenter, right ,AD = AC,BE = BC,AP = AE,BQ = BD (Argentina 2016 OMA L1 p3)
parmenides51   2
N Feb 1, 2025 by sunken rock
Let $ABC$ be a right triangle. Points $D$ and $E$ on hypotenuse $AB$ are such that $AD = AC$ and $BE = BC$. Points $P$ and $Q$ on sides $AC$ and $BC$ respectively are such that $AP = AE$ and $BQ = BD$. Let $M$ be the midpoint of $PQ$.
[list=a]
[*]Prove that $M$ is the intersection point of the angle bisectors of triangle $ABC$.
[*]Determine the measure of angle $\angle AMB$.
[/list]
2 replies
parmenides51
May 15, 2020
sunken rock
Feb 1, 2025
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2014 Mock Western PA ARML Team #10 area of triangle by feet of angle bisectors
parmenides51   2
N Jan 31, 2025 by KSH31415
In $\vartriangle ABC$ with side lengths $3$, $4$, and $5$, angle bisectors $AX$, $BY$ , and $CZ$ are drawn, where $X$ lies on $BC$, $Y$ lies on $AC$, and $Z$ lies on $AB$. Compute the area of $\vartriangle XY Z$.
2 replies
parmenides51
Jan 21, 2024
KSH31415
Jan 31, 2025
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angle bisector // MN, BD = CE. midpoints (Argentina 2019 OMA L1 p3)
parmenides51   4
N Jan 29, 2025 by sunken rock
In triangle $ABC$ let $D$ and $E$ be on sides $AB$ and $AC$, respectively, such that $BD = CE$. Let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively. Prove that the bisector of the angle $\angle BAC$ is parallel to the line $MN$.
4 replies
parmenides51
May 15, 2020
sunken rock
Jan 29, 2025
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J
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Easy number theory problem
ilikemath247365   9
N Jun 7, 2025 by ilikemath247365
Find two consecutive three digit numbers such that if you take the largest prime factor for each of these numbers and add them, you get $150$.
9 replies
ilikemath247365
Jun 4, 2025
ilikemath247365
Jun 7, 2025
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Easy number theory problem
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Reveal topic tags
number theory
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ilikemath247365
298 posts
ilikemath247365
#1 h  Jun 4, 2025, 1:54 PM
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Find two consecutive three digit numbers such that if you take the largest prime factor for each of these numbers and add them, you get $150$.
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Oshawoot
157 posts
Oshawoot
#2 h  Jun 4, 2025, 2:30 PM • 1 Y
Y by ilikemath247365
answer but not really sol
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ilikemath247365
298 posts
ilikemath247365
#3 h  Jun 4, 2025, 2:32 PM
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Nice this works. See if you can find all pairs of consecutive three digit numbers that satisfy the above conditions.
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deeptisidana
41 posts
deeptisidana
#4 h  Jun 4, 2025, 2:33 PM • 1 Y
Y by ilikemath247365
I don’t know how to hide but my solution is you plug in numbers into 150 so it’s 71 and 79 for the factors and I used a calculator to find the value 710, 711 which respectively is 71*10, 79*9. This solution is not time bound and is used with a calcula
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deeptisidana
41 posts
deeptisidana
#5 h  Jun 4, 2025, 2:33 PM
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Calculator
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Oshawoot
157 posts
Oshawoot
#6 h  Jun 4, 2025, 2:36 PM • 1 Y
Y by ilikemath247365
deeptisidana wrote:
I don’t know how to hide but my solution is you plug in numbers into 150 so it’s 71 and 79 for the factors and I used a calculator to find the value 710, 711 which respectively is 71*10, 79*9. This solution is not time bound and is used with a calcula

to hide on computer there's a button thats an eye crossed out, highlight your text and click it
on the other hand if you cant do that just type [hide.][/hide] without the period in the first one and put your text in the middle

also that solution's basically what i did too!
deeptisidana wrote:
Calculator
you can just edit your old post instead of making a new one, there's a pencil button next to your posts you can use to edit
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deeptisidana
41 posts
deeptisidana
#7 h  Jun 4, 2025, 2:37 PM • 1 Y
Y by ilikemath247365
Click to reveal hidden text
This post has been edited 1 time. Last edited by deeptisidana, Jun 4, 2025, 2:39 PM
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NamelyOrange
540 posts
NamelyOrange
#8 h  Jun 4, 2025, 2:37 PM • 1 Y
Y by ilikemath247365
Reworded question: let $\operatorname{Gpf}(n)$ be the greatest prime factor of $n$. Find all three digit $n$ such that $\operatorname{Gpf}(n)+\operatorname{Gpf}(n+1) = 150$.

brute force
This post has been edited 6 times. Last edited by NamelyOrange, Jun 7, 2025, 12:38 AM
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NamelyOrange
540 posts
NamelyOrange
#9 h  Jun 7, 2025, 12:39 AM
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I finished bashing. Are these all the solutions?

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This post has been edited 1 time. Last edited by NamelyOrange, Jun 7, 2025, 3:49 AM

				

			

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ilikemath247365 					

					

						#11
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						Jun 7, 2025, 2:48 AM
						
						
					

				

				
Y by 
Yes awesome job! These are all the solutions every single one of them according to my work. Great job!

					

					

				

			

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