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

Plan ahead for the next school year. Schedule your class today!
derivative
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
[Sipnayan 2025 JHS Eliminations] Easy 1.5
pensive   1
N Yesterday at 11:21 AM by pensive
Let $20, a_1, a_2, a_3, \ldots$ and $25, b_1, b_2, b_3, \ldots$ be geometric sequences with the same common ratio. It is given that $a_{2023} = b_{2025}$. If $\dfrac{a_2 + a_4 + a_6 + a_8 + \ldots}{b_2 - b_4 + b_6 - b_8 + \ldots} = \dfrac{p}{q}$ where $p, q$ are relatively prime positive integers, find $p + q$.

Answer
1 reply
pensive
Yesterday at 11:20 AM
pensive
Yesterday at 11:21 AM
J
H
4 secants forming a quadrilateral
pensive   1
N Jul 7, 2025 by pensive
In the figure, points $A$, $B$, $C$, and $D$ all lie on the circle. Meanwhile, $CD = DE$, $\angle E = 48^\circ$, and $\overset{\frown}{BC}$ and $\overset{\frown}{AD}$ are in the ratio 2 : 5. What's the measure of minor arc $\overset{\frown}{BC}$?

Answer
1 reply
pensive
Jul 7, 2025
pensive
Jul 7, 2025
J
H
geometry
wpdnjs   1
N Jun 29, 2025 by Zusen
A circle is inscribed in a regular hexagon and another cirlce is circumscribed about the hexagon. Find the ratio of the area of the smaller circle to the area of the larger circle.
1 reply
wpdnjs
Jun 29, 2025
Zusen
Jun 29, 2025
J
H
Choosing a Class Representative
ernie   6
N Jun 29, 2025 by nudinhtien
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is

$ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$
6 replies
ernie
Mar 2, 2009
nudinhtien
Jun 29, 2025
J
H
Trisecting Cevians
Brut3Forc3   8
N Jun 28, 2025 by SomeonecoolLovesMaths
In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF.$ Then the area of triangle $ N_1N_2N_3$ is:
IMAGE$ \textbf{(A)}\ \frac {1}{10} \triangle ABC \qquad\textbf{(B)}\ \frac {1}{9} \triangle ABC \qquad\textbf{(C)}\ \frac {1}{7} \triangle ABC \qquad\textbf{(D)}\ \frac {1}{6} \triangle ABC \qquad\textbf{(E)}\ \text{none of these}$
8 replies
Brut3Forc3
Feb 3, 2009
SomeonecoolLovesMaths
Jun 28, 2025
J
H
2017 preRMO p12, boys : girls = 4:3, absent 8 boys and 14 girls
parmenides51   2
N Jun 24, 2025 by cortex_classes
In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
2 replies
parmenides51
Aug 9, 2019
cortex_classes
Jun 24, 2025
J
H
Finding area of a right triangle given its ratio
Kyj9981   1
N Jun 11, 2025 by Kyj9981
Source: 19th PMO Area Stage Part I Problem #19

The lengths of the two legs of a right triangle are in the ratio of $7 : 24$. The distance between its
incenter and its circumcenter is $1$. Find its area.

Answer
1 reply
Kyj9981
Jun 11, 2025
Kyj9981
Jun 11, 2025
J
H
2014 preRMO p10, computational with ratios and areas
parmenides51   11
N May 16, 2025 by MATHS_ENTUSIAST
In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?
11 replies
parmenides51
Aug 9, 2019
MATHS_ENTUSIAST
May 16, 2025
J
H
Malaysia MO IDM UiTM 2025
smartvong   2
N May 11, 2025 by Sivin
MO IDM UiTM 2025 (Category C)

Contest Description

Preliminary Round
Section A
1. Given that $2^a + 2^b = 2016$ such that $a, b \in \mathbb{N}$. Find the value of $a$ and $b$.

2. Find the value of $a, b$ and $c$ such that $$\frac{ab}{a + b} = 1, \frac{bc}{b + c} = 2, \frac{ca}{c + a} = 3.$$
3. If the value of $x + \dfrac{1}{x}$ is $\sqrt{3}$, then find the value of
$$x^{1000} + \frac{1}{x^{1000}}$$.

Section B
1. Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integer $a, b$:
$$f(2a) + 2f(b) = f(f(a + b))$$
2. The side lengths $a, b, c$ of a triangle $\triangle ABC$ are positive integers. Let
$$T_n = (a + b + c)^{2n} - (a - b + c)^{2n} - (a + b - c)^{2n} - (a - b - c)^{2n}$$for any positive integer $n$.
If $\dfrac{T_2}{2T_1} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $\triangle ABC$.

Final Round
Section A
1. Given that the equation $x^2 + (b - 3)x - 2b^2 + 6b - 4 = 0$ has two roots, where one is twice of the other, find all possible values of $b$.

2. Let $$f(y) = \dfrac{y^2}{y^2 + 1}.$$Find the value of $$f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + \cdots + f\left(\frac{2000}{2001}\right) + f\left(\frac{2001}{2001}\right) + f\left(\frac{2001}{2000}\right) + \cdots + f\left(\frac{2001}{2}\right) + f\left(\frac{2001}{1}\right).$$
3. Find the smallest four-digit positive integer $L$ such that $\sqrt{3\sqrt{L}}$ is an integer.

Section B
1. Given that $\tan A : \tan B : \tan C$ is $1 : 2 : 3$ in triangle $\triangle ABC$, find the ratio of the side length $AC$ to the side length $AB$.

2. Prove that $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$
2 replies
smartvong
May 10, 2025
Sivin
May 11, 2025
J
H
Pythagorean triples vs sine ratio?
Miranda2829   6
N Apr 10, 2025 by anticodon
I'm a bit confused about the

right angle 3 4 5 have a sine ratio of 0.6 and cosine of 0.8,

Do different lengths of right-angle triangles have different ratios?

how to get an actual angle of sine ?

thanks

6 replies
Miranda2829
Feb 27, 2025
anticodon
Apr 10, 2025
J
a