Question from Gazeta matematica

by abcdefghijklmop, May 10, 2025, 7:30 PM

Determine how many subsets formed by 7 elements which are in geometric progession are in the set
{1,2,....,2025}.

Nice problem

by gasgous, May 10, 2025, 1:30 PM

Given that the product of three integers is $60$ .What is the least possible positive sum of the three integers?

Malaysia MO IDM UiTM 2025

by smartvong, May 10, 2025, 1:01 PM

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 $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$

This post has been edited 1 time. Last edited by smartvong, Yesterday at 1:02 PM

function

AIME

trigonometry

ratio

Inequalities

by sqing, May 10, 2025, 12:50 PM

Let $a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{38}{23}$ Let $a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2(2\sqrt{3}+1)}{5}$

inequalities

Angle Formed by Points on the Sides of a Triangle

by xeroxia, May 10, 2025, 10:28 AM

In triangle $ABC$ , points $D$ , $E$ , and $F$ lie on sides $BC$ , $CA$ , and $AB$ , respectively, such that
$BD = 20$ , $DC = 15$ , $CE = 13$ , $EA = 8$ , $AF = 6$ , $FB = 22$ .

What is the measure of $\angle EDF$ ?

geometry

Calculate the distance AD

by MTA_2024, May 9, 2025, 3:50 PM

A semi-circle is inscribed in a quadrilateral $ABCD$ . The center $O$ of the semi-circle is the midpoint of segment $AD$ . We have $AB=9$ and $CD=16$ .
Calculate the distance $AD$ .

Attachments:

geometry

circle

Range if \omega for No Inscribed Right Triangle y = \sin(\omega x)

by ThisIsJoe, May 8, 2025, 2:02 PM

For a positive number \omega , determine the range of \omega for which the curve y = \sin(\omega x) has no inscribed right triangle.
Could someone help me figure out how to approach this?

Unknown triangle area

by smartvong, May 8, 2025, 1:38 AM

The diagram shows a convex quadrilateral $ABCD$ . The points $E$ and $F$ divide $AB$ into three equal parts while the points $G$ and $H$ divide $CD$ into three equal parts. The line segments $AH$ and $ED$ intersect at $I$ . The line segments $CF$ and $BG$ intersect at $J$ . Given that the areas of the triangles $AID$ , $EHI$ and $FJG$ are $154$ , $112$ , and $99$ respectively, find the area of the triangle $BJC$ .

$[asy] import olympiad; // for marksegment size(300); pair A = (1,4); pair B = (7,4.5); pair C = (6.5,2); pair D = (0,1); // trisect AB pair E = A + (B - A)/3; pair F = A + 2*(B - A)/3; // trisect CD pair G = C + (D - C)/3; pair H = C + 2*(D - C)/3; // draw the quadrilateral draw(A--B--C--D--cycle); // draw the interior segments draw(A--H); draw(D--E); draw(C--F); draw(B--G); // draw the “verticals” EH and FG draw(E--H); draw(F--G); // intersections pair I = intersectionpoint(A--H, D--E); pair J = intersectionpoint(C--F, B--G); // dots & labels dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); label("A", A, NW); label("B", B, NE); label("C", C, SE); label("D", D, SW); label("E", E, N); label("F", F, N); label("G", G, S); label("H", H, S); label("I", I, W); label("J", J, W); // triangle‐center labels label("154", (A + I + D)/3); label("112", (E + I + H)/3); label("99", (F + J + G)/3); // congruence tick marks // AE = EF = FB (single tick) add(pathticks(A--E, 1)); add(pathticks(E--F, 1)); add(pathticks(F--B, 1)); // DH = HG = GC (double tick) add(pathticks(D--H, 2)); add(pathticks(H--G, 2)); add(pathticks(G--C, 2)); [/asy]$

This post has been edited 1 time. Last edited by smartvong, May 8, 2025, 1:38 AM

Triangles

geometry

area

Interesting question from Al-Khwarezmi olympiad 2024 P3, day1

by Adventure1000, May 7, 2025, 4:10 PM

Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$\begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases}$ Proposed by Ngo Van Trang, Vietnam

inequalities

2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0

by parmenides51, Jun 17, 2019, 12:42 PM

Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$ . What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?

Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
lol⠀⠀⠀⠀⠀
by math_explorer, Jan 20, 2021, 8:43 AM
woah ancient blog
by suvamkonar, Jan 20, 2021, 4:14 AM
https://artofproblemsolving.com/community/c47h361466
by math_explorer, Jun 10, 2020, 1:20 AM
when did the first greed control game start?
by piphi, May 30, 2020, 1:08 AM
ok..........
by asdf334, Sep 10, 2019, 3:48 PM
There is one existing way to obtain contributorship documented on this blog. See if you can find it.
by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
Hullo bye
by AnArtist, Jan 15, 2019, 8:59 AM
Hullo bye
by tastymath75025, Nov 22, 2018, 9:08 PM
Hullo bye
by Kayak, Jul 22, 2018, 1:29 PM
It's sad; the blog is still active but not really ;-;
by GeneralCobra19, Sep 21, 2017, 1:09 AM
dope css
by zxcv1337, Mar 27, 2017, 4:44 AM
nice blog ^_^
by chezbgone, Mar 28, 2016, 5:18 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:58 PM
I like your CSS. Can you send me your shoutbox block? I'll credit you.
by bluecarneal, Feb 21, 2011, 8:24 PM
Hey contrib?
by Dsquared, Dec 16, 2010, 3:14 PM
$\sum_{n=0}^{\infty}2^n=-1$
by AwesomeToad, Dec 13, 2010, 2:12 PM
Delete the shouts...

If you wish something more mathematical, here's some digits.
1679350278936041982752
Enjoy. (no offense)
by chaotic_iak, Dec 3, 2010, 1:00 PM
Now I feel guilty my shoutbox is so nonmathematical.
by math_explorer, Dec 3, 2010, 12:36 PM
Congratulations on winning Achievement Unlocked: AoPS Style! I'm looking to your participation in Achievement Unlocked 2: Reunlock the Achievements!
by chaotic_iak, Dec 3, 2010, 12:32 PM
El_ectric won game 11
by halowarfare17, Nov 17, 2010, 2:36 PM
Did you win reaper or did electric?
by dragon96, Nov 11, 2010, 2:14 AM
Very nice blog
by Amir Hossein, Oct 28, 2010, 2:58 PM
Changed your avatar?
by asf, Oct 15, 2010, 3:47 AM
Gosh, post count over 9000.
by math_explorer, Sep 20, 2010, 1:44 PM
Nice blog I read pretty much whenever you make a new post, but I don't comment.
by AIME15, Sep 19, 2010, 2:38 PM
I see, that part of the CSS needs a lot of tweaking.
by math_explorer, Sep 12, 2010, 9:20 AM
Hello. Shout?
by asf, Sep 11, 2010, 5:18 PM