Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
geometry
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Coaxal Circles
fattypiggy123   30
N 15 minutes ago by Ilikeminecraft
Source: China TSTST Test 2 Day 1 Q3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
30 replies
fattypiggy123
Mar 13, 2017
Ilikeminecraft
15 minutes ago
J
H
Inequalities
sqing   3
N 5 hours ago by sqing
Let $a,b,c >2 $ and $ ab+bc+ca \leq 75.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$$Let $a,b,c >2 $ and $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$$
3 replies
sqing
Yesterday at 11:31 AM
sqing
5 hours ago
J
H
Assam Mathematics Olympiad 2022 Category III Q18
SomeonecoolLovesMaths   2
N 6 hours ago by nyacide
Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that
(a) $ f(m) < f(n)$ whenever $m < n$.
(b) $f(2n) = f(n) + n$ for all $n \in \mathbb{N}$.
(c) $n$ is prime whenever $f(n)$ is prime.
Find $$\sum_{n=1}^{2022} f(n).$$
2 replies
SomeonecoolLovesMaths
Sep 12, 2024
nyacide
6 hours ago
J
H
Assam Mathematics Olympiad 2022 Category III Q17
SomeonecoolLovesMaths   1
N Today at 7:24 AM by nyacide
Consider a rectangular grid of points consisting of $4$ rows and $84$ columns. Each point is coloured with one of the colours red, blue or green. Show that no matter whatever way the colouring is done, there always exist four points
of the same colour that form the vertices of a rectangle. An illustration is shown in the figure below.
1 reply
SomeonecoolLovesMaths
Sep 12, 2024
nyacide
Today at 7:24 AM
J
H
Assam Mathematics Olympiad 2022 Category III Q14
SomeonecoolLovesMaths   1
N Today at 6:54 AM by nyacide
The following sum of three four digits numbers is divisible by $75$, $7a71 + 73b7 + c232$, where $a, b, c$ are decimal digits. Find the necessary conditions in $a, b, c$.
1 reply
SomeonecoolLovesMaths
Sep 12, 2024
nyacide
Today at 6:54 AM
J
H
Assam Mathematics Olympiad 2022 Category III Q12
SomeonecoolLovesMaths   2
N Today at 6:20 AM by nyacide
A particle is in the origin of the Cartesian plane. In each step the particle can go $1$ unit in any of the directions, left, right, up or down. Find the number of ways to go from $(0, 0)$ to $(0, 2)$ in $6$ steps. (Note: Two paths where identical set of points is traversed are considered different if the order of traversal of each point is different in both paths.)
2 replies
SomeonecoolLovesMaths
Sep 12, 2024
nyacide
Today at 6:20 AM
J
H
Assam Mathematics Olympiad 2022 Category III Q10
SomeonecoolLovesMaths   1
N Today at 5:53 AM by nyacide
Let the vertices of the square $ABCD$ are on a circle of radius $r$ and with center $O$. Let $P, Q, R$ and $S$ are the mid points of $AB, BC, CD$ and $DA$ respectively. Then;
(a) Show that the quadrilateral $P QRS$ is a square.
(b) Find the distance from the mid point of $P Q$ to $O$.
1 reply
SomeonecoolLovesMaths
Sep 12, 2024
nyacide
Today at 5:53 AM
J
H
A problem of collinearity.
Raul_S_Baz   2
N Today at 4:11 AM by Raul_S_Baz
Î am the author.
IMAGE
P.S: How can I verify that it is an original problem? Thanks!
2 replies
Raul_S_Baz
Yesterday at 4:19 PM
Raul_S_Baz
Today at 4:11 AM
J
H
Inequalities
sqing   0
Today at 3:46 AM
Let $ a,b,c>0 $ . Prove that
$$\frac{a+5b}{b+c}+\frac{b+5c}{c+a}+\frac{c+5a}{a+b}\geq 9$$$$ \frac{2a+11b}{b+c}+\frac{2b+11c}{c+a}+\frac{2c+11a}{a+b}\geq \frac{39}{2}$$$$ \frac{25a+147b}{b+c}+\frac{25b+147c}{c+a}+\frac{25c+147a}{a+b} \geq258$$
0 replies
sqing
Today at 3:46 AM
0 replies
J
H
Plz help
Bet667   3
N Yesterday at 6:50 PM by K1mchi_
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)
3 replies
Bet667
Jan 28, 2024
K1mchi_
Yesterday at 6:50 PM
J
H
2019 SMT Team Round - Stanford Math Tournament
parmenides51   17
N Yesterday at 6:40 PM by Rombo
p1. Given $x + y = 7$, find the value of x that minimizes $4x^2 + 12xy + 9y^2$.


p2. There are real numbers $b$ and $c$ such that the only $x$-intercept of $8y = x^2 + bx + c$ equals its $y$-intercept. Compute $b + c$.



p3. Consider the set of $5$ digit numbers $ABCDE$ (with $A \ne 0$) such that $A+B = C$, $B+C = D$, and $C + D = E$. What’s the size of this set?


p4. Let $D$ be the midpoint of $BC$ in $\vartriangle ABC$. A line perpendicular to D intersects $AB$ at $E$. If the area of $\vartriangle ABC$ is four times that of the area of $\vartriangle BDE$, what is $\angle ACB$ in degrees?


p5. Define the sequence $c_0, c_1, ...$ with $c_0 = 2$ and $c_k = 8c_{k-1} + 5$ for $k > 0$. Find $\lim_{k \to \infty} \frac{c_k}{8^k}$.


p6. Find the maximum possible value of $|\sqrt{n^2 + 4n + 5} - \sqrt{n^2 + 2n + 5}|$.


p7. Let $f(x) = \sin^8 (x) + \cos^8(x) + \frac38 \sin^4 (2x)$. Let $f^{(n)}$ (x) be the $n$th derivative of $f$. What is the largest integer $a$ such that $2^a$ divides $f^{(2020)}(15^o)$?


p8. Let $R^n$ be the set of vectors $(x_1, x_2, ..., x_n)$ where $x_1, x_2,..., x_n$ are all real numbers. Let $||(x_1, . . . , x_n)||$ denote $\sqrt{x^2_1 +... + x^2_n}$. Let $S$ be the set in $R^9$ given by $$S = \{(x, y, z) : x, y, z \in R^3 , 1 = ||x|| = ||y - x|| = ||z -y||\}.$$If a point $(x, y, z)$ is uniformly at random from $S$, what is $E[||z||^2]$?


p9. Let $f(x)$ be the unique integer between $0$ and $x - 1$, inclusive, that is equivalent modulo $x$ to $\left( \sum^2_{i=0} {{x-1} \choose i} ((x - 1 - i)! + i!) \right)$. Let $S$ be the set of primes between $3$ and $30$, inclusive. Find $\sum_{x\in S}^{f(x)}$.


p10. In the Cartesian plane, consider a box with vertices $(0, 0)$,$\left( \frac{22}{7}, 0\right)$,$(0, 24)$,$\left( \frac{22}{7}, 4\right)$. We pick an integer $a$ between $1$ and $24$, inclusive, uniformly at random. We shoot a puck from $(0, 0)$ in the direction of $\left( \frac{22}{7}, a\right)$ and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at $(0, 0)$ and when it ends at some vertex of the box?


p11. Sarah is buying school supplies and she has $\$2019$. She can only buy full packs of each of the following items. A pack of pens is $\$4$, a pack of pencils is $\$3$, and any type of notebook or stapler is $\$1$. Sarah buys at least $1$ pack of pencils. She will either buy $1$ stapler or no stapler. She will buy at most $3$ college-ruled notebooks and at most $2$ graph paper notebooks. How many ways can she buy school supplies?


p12. Let $O$ be the center of the circumcircle of right triangle $ABC$ with $\angle ACB = 90^o$. Let $M$ be the midpoint of minor arc $AC$ and let $N$ be a point on line $BC$ such that $MN \perp BC$. Let $P$ be the intersection of line $AN$ and the Circle $O$ and let $Q$ be the intersection of line $BP$ and $MN$. If $QN = 2$ and $BN = 8$, compute the radius of the Circle $O$.


p13. Reduce the following expression to a simplified rational $$\frac{1}{1 - \cos \frac{\pi}{9}}+\frac{1}{1 - \cos \frac{5 \pi}{9}}+\frac{1}{1 - \cos \frac{7 \pi}{9}}$$

p14. Compute the following integral $\int_0^{\infty} \log (1 + e^{-t})dt$.


p15. Define $f(n)$ to be the maximum possible least-common-multiple of any sequence of positive integers which sum to $n$. Find the sum of all possible odd $f(n)$


PS. You should use hide for answers. Collected here.
17 replies
parmenides51
Feb 6, 2022
Rombo
Yesterday at 6:40 PM
J
H
J
H
Trivial fun Equilateral
ItzsleepyXD   5
N May 1, 2025 by reni_wee
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
5 replies
ItzsleepyXD
Apr 30, 2025
reni_wee
May 1, 2025
J
Trivial fun Equilateral
G H J
Reveal topic tags
geometry
Equilateral
L
G H BBookmark kLocked kLocked NReply
Source: Own , Mock Thailand Mathematic Olympiad P1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ItzsleepyXD
149 posts
ItzsleepyXD
#1 h  Apr 30, 2025, 9:05 AM
Y by
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
moony_
22 posts
moony_
#2 h  Apr 30, 2025, 11:06 AM • 1 Y
Y by Retemoeg
PQ, XW, YZ are concurrent cuz pappus theorem for points X, B, Z and Y, C, W
=> Desargues theorem for triangles XPY and WQZ and YP || ZQ, XP || WQ => ZW || XY too

cool problem >w<
This post has been edited 1 time. Last edited by moony_, Apr 30, 2025, 2:52 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
moony_
22 posts
moony_
#3 h  Apr 30, 2025, 11:08 AM
Y by
generalization btw: BPCQ - parallelogram
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Tsikaloudakis
981 posts
Tsikaloudakis
#4 h  Apr 30, 2025, 12:07 PM
Y by
apothikefsi
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
cursed_tangent1434
634 posts
cursed_tangent1434
#5 h  May 1, 2025, 6:01 AM • 1 Y
Y by reni_wee
Alternative solution via trigonometry. We WLOG consider the case where $ABC$ is obtuse with $AC > AB> AC$ to avoid configurational confusion. Now, note that by the Law of Sines on $\triangle AWB$,
\[\frac{AW}{AB} = \frac{\sin \angle ABW}{\sin \angle AWB} = \frac{\sin (120-B)}{ \sin (60-C)}\]Similarly,
\[\frac{AZ}{AC} = \frac{\sin (60+C)}{\sin (B-60)}\]\[\frac{AY}{AB} = \frac{\sin (B-60)}{\sin (C+60)}\]and
\[\frac{AX}{AC} = \frac{\sin (60-C)}{\sin (120-B)}\]Now it is easy to see that,
\[\frac{AW}{AZ} = \frac{\sin (120-B)}{ \sin (60-C)} \cdot \frac{\sin (B-60)}{\sin (60+C)} \cdot \frac{AB}{AC}\]and
\[\frac{AY}{AX} = \frac{\sin (B-60)}{\sin (C+60)} \cdot \frac{\sin (120-B)}{\sin (60-C)} \cdot \frac{AB}{AC}\]from which we may conclude that
\[\frac{AW}{AZ} = \frac{AY}{AX}\]implying that $XY \parallel WZ$ as desired.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
reni_wee
48 posts
reni_wee
#6 h  May 1, 2025, 5:50 PM • 1 Y
Y by cursed_tangent1434
Easy Problem. Barely MOHS 5.
WLOG Let $AC > AB$. As $\triangle BPC$ and $\triangle BQC$ are equilateral, $BP \parallel QC, PC \parallel BQ \implies \triangle AYB \sim \triangle ACZ, \triangle XAC \sim \triangle BAW$.
Let us handle these 2 separately. We get 2 equations as follows,
$$\frac{AB}{AZ} = \frac {AY}{AC} \implies AZ \cdot AY = AB \cdot AC$$$$\frac{AW}{AC} = \frac {AB}{AX} \implies AW \cdot AX = AB \cdot AC$$$\implies AW \cdot AX =AZ \cdot AY \implies \triangle XAY \sim \triangle ZAW$
From which we can get that $XY \parallel WZ$
Attachments:
This post has been edited 1 time. Last edited by reni_wee, May 1, 2025, 7:55 PM
Z K Y
N Quick Reply
G
H
=
a