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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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0 replies
jlacosta
May 1, 2025
0 replies
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Factorials divide
va2010   37
N 22 minutes ago by ND_
Source: 2015 ISL N2
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
37 replies
va2010
Jul 7, 2016
ND_
22 minutes ago
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IMO Shortlist 2011, Number Theory 2
orl   24
N 25 minutes ago by ezpotd
Source: IMO Shortlist 2011, Number Theory 2
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.

Proposed by Luxembourg
24 replies
orl
Jul 11, 2012
ezpotd
25 minutes ago
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Inequality in triangle
Nguyenhuyen_AG   3
N 33 minutes ago by Nguyenhuyen_AG
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]
3 replies
Nguyenhuyen_AG
Today at 6:17 AM
Nguyenhuyen_AG
33 minutes ago
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Problem 1
randomusername   73
N 44 minutes ago by ND_
Source: IMO 2015, Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Proposed by Netherlands
73 replies
randomusername
Jul 10, 2015
ND_
44 minutes ago
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Is problem true?!?!?!?
giangtruong13   0
Yesterday at 2:00 PM
Let $ABC$ be a triangle, $I$ is incenter of triangle $ABC$. Draw $IM$ perpendicular to $AB$ at $M$ and $IN$ perpendicular to $AC$ at $N$, $IM=IN=m$. Prove that: Area of triangle $ANM$ $\geq 2m^2$
0 replies
giangtruong13
Yesterday at 2:00 PM
0 replies
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rare creative geo problem spotted in the wild
abbominable_sn0wman   4
N Yesterday at 11:21 AM by abbominable_sn0wman
The following is the construction of the twindragon fractal.

Let $I_0$ be the solid square region with vertices at
\[ (0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right). \]
Recursively, the region $I_{n+1}$ consists of two copies of $I_n$: one copy which is rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and another copy which is also rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and then translated by $\left(\frac{1}{2}, -\frac{1}{2}\right)$.

We have displayed $I_0$ and $I_1$ below.

Let $I_\infty$ be the limiting region of the sequence $I_0, I_1, \dots$.

The area of the smallest convex polygon which encloses $I_\infty$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a + b$.
4 replies
abbominable_sn0wman
Thursday at 6:04 PM
abbominable_sn0wman
Yesterday at 11:21 AM
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Indonesia Juniors 2012 day 2 OSN SMP
parmenides51   3
N Yesterday at 11:18 AM by Rayholr123
p1. One day, a researcher placed two groups of species that were different, namely amoeba and bacteria in the same medium, each in a certain amount (in unit cells). The researcher observed that on the next day, which is the second day, it turns out that every cell species divide into two cells. On the same day every cell amoeba prey on exactly one bacterial cell. The next observation carried out every day shows the same pattern, that is, each cell species divides into two cells and then each cell amoeba prey on exactly one bacterial cell. Observation on day $100$ shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day.


p2. It is known that $n$ is a positive integer. Let $f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$.
Find $f(13) + f(14) + f(15) + ...+ f(112).$


p3. Budi arranges fourteen balls, each with a radius of $10$ cm. The first nine balls are placed on the table so that
form a square and touch each other. The next four balls placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four balls. If Bambang has fifty five balls each also has a radius of $10$ cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang.


p4. Given a triangle $ABC$ whose sides are $5$ cm, $ 8$ cm, and $\sqrt{41}$ cm. Find the maximum possible area of the rectangle can be made in the triangle $ABC$.


p5. There are $12$ people waiting in line to buy tickets to a show with the price of one ticket is $5,000.00$ Rp.. Known $5$ of them they only have $10,000$ Rp. in banknotes and the rest is only has a banknote of $5,000.00$ Rp. If the ticket seller initially only has $5,000.00$ Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?
3 replies
parmenides51
Nov 3, 2021
Rayholr123
Yesterday at 11:18 AM
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Indonesian Junior MO (Nationals) 2018, Day 2
somebodyyouusedtoknow   1
N Yesterday at 10:12 AM by Rayholr123
P6. It is given the integer $Y$ with
$Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$
Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.)

P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....

P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$.

P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal.

(Image should be placed here, look at attachment.)

a) Determine the position of the number $2018$ based on its row, column, and diagonal.
b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$.

P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a gadang number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a gadang number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a gadang number?
1 reply
somebodyyouusedtoknow
Nov 11, 2021
Rayholr123
Yesterday at 10:12 AM
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Indonesian Junior MO 2018 (Nationals), Day 1
somebodyyouusedtoknow   6
N Yesterday at 10:07 AM by Rayholr123
The problems are really difficult to find online, so here are the problems.

P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form.
a) Determine the smallest possible value of $k$.
b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$.
c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.

I added this because my translation is a bit weird.
Indonesian Version

P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs.
Graph Construction Notes
Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$.
a) Draw the graph of the function $g \circ f$.
b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$.

P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$.

P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$.

P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.

6 replies
somebodyyouusedtoknow
Nov 11, 2021
Rayholr123
Yesterday at 10:07 AM
J
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Great Geometry with Squares on sides of triangles
SomeonecoolLovesMaths   4
N Yesterday at 4:50 AM by ohiorizzler1434
Three squares are drawn on the sides of triangle \(ABC\) (i.e., the square on \(AB\) has \(AB\) as one of its sides and lies outside \(ABC\)). Show that the lines drawn from the vertices \(A\), \(B\), and \(C\) to the centers of the opposite squares are concurrent.

IMAGE
4 replies
SomeonecoolLovesMaths
May 22, 2025
ohiorizzler1434
Yesterday at 4:50 AM
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9 Isogonal and isotomic conjugates
V0305   13
N Yesterday at 2:32 AM by ohiorizzler1434
1. Do you think isogonal conjugates should be renamed to angular conjugates?
2. Do you think isotomic conjugates should be renamed to cevian conjugates?

Please answer truthfully :)

Credit to Stead for this renaming idea
13 replies
V0305
May 26, 2025
ohiorizzler1434
Yesterday at 2:32 AM
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Interesting Geometry
captainmath99   4
N Thursday at 8:01 PM by captainmath99
Let ABC be a right triangle such that $\angle{C}=90^\circ, CA=6, CB=4$. A circle O with center C has a radius of 2. Let P be a point on the circle O.

a)What is the minimum value of $(AP+\dfrac{1}{2}BP)$?
Answer Check

b) What is the minimum value of $(\dfrac{1}{3}AP+BP)$?
Answer Check
4 replies
captainmath99
May 25, 2025
captainmath99
Thursday at 8:01 PM
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Geometry
AlexCenteno2007   1
N May 28, 2025 by ohiorizzler1434
Given triangle ABC, it is true that BD = CF where D and F are points in the same half-plane with respect to line BC and it is also known that BD is parallel to AC and CF is parallel to AB. Show that BF, CD and the interior bisector of A are concurrent.
1 reply
AlexCenteno2007
May 28, 2025
ohiorizzler1434
May 28, 2025
J
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21st PMO National Orals #9
yes45   0
May 28, 2025
In square $ABCD$, $P$ and $Q$ are points on sides $CD$ and $BC$, respectively, such that $\angle{APQ} = 90^\circ$. If $AP = 4$ and $PQ = 3$, find the area of $ABCD$.

Answer Confirmation
Solution
0 replies
yes45
May 28, 2025
0 replies
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J
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Tetrahedron and equality of segments
Megus   3
N Dec 12, 2019 by soryn
$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$. Show that $AB = CD$.
3 replies
Megus
Oct 30, 2005
soryn
Dec 12, 2019
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Tetrahedron and equality of segments
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Reveal topic tags
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Megus
1198 posts
Megus
#1 h  Oct 30, 2005, 2:22 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$. Show that $AB = CD$.
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soryn
5348 posts
soryn
#3 h  Dec 12, 2019, 7:01 AM • 2 Y
Y by Adventure10, Mango247
Any ideas, pls??
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Hax.Mode.ON3
39 posts
Hax.Mode.ON3
#4 h  Dec 12, 2019, 8:30 AM • 2 Y
Y by Adventure10, Mango247
soryn wrote:
Any ideas, pls??

just open the tetrahedron and it make it 2D . then its really easy.
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soryn
5348 posts
soryn
#5 h  Dec 12, 2019, 9:35 PM • 1 Y
Y by Adventure10
Yes.......
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