.problem.

by Cobedangiu, Apr 4, 2025, 6:20 AM

Find the integer coefficients after expanding Newton's binomial:
$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$

This post has been edited 1 time. Last edited by Cobedangiu, Yesterday at 6:20 AM

algebra

New geometry problem

by titaniumfalcon, Apr 3, 2025, 10:40 PM

Post any solutions you have, with explanation or proof if possible, good luck!

Attachments:

geometry

Excalibur Identity

by jjsunpu, Apr 3, 2025, 3:27 PM

proof is below

Attachments:

This post has been edited 1 time. Last edited by jjsunpu, Yesterday at 12:19 AM
Reason: idk

algebraic identities

Probability

by Ecrin_eren, Apr 3, 2025, 11:21 AM

In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

combinatorics

probability

Congruence

by Ecrin_eren, Apr 3, 2025, 10:34 AM

Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

number theory

School Math Problem

by math_cool123, Apr 2, 2025, 5:03 AM

Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $(a^2+b)(a+b^2)=(a-b)^3.$

Inequalities

by sqing, Apr 1, 2025, 2:59 PM

Let $a, b,c\geq 0$ and $2a+3b+ 4c=11.$ Prove that
$a+ab+abc\leq\frac{49}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=10.$ Prove that
$a+ab+abc\leq\frac{169}{24}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=14.$ Prove that
$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=32.$ Prove that
$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$

inequalities

Geo Mock #10

by Bluesoul, Apr 1, 2025, 7:06 AM

Consider acute $\triangle{ABC}$ with $AB=10$ , $AC<BC$ and area $135$ . The circle $\omega$ with diameter $AB$ meets $BC$ at $E$ . Let the orthocenter of the triangle be $H$ , connect $CH$ and extend to meet $\omega$ at $N$ such that $NC>HC$ and $NE$ is the diameter of $\omega$ . Draw the circumcircle $\Gamma$ of $\triangle{AHB}$ , chord $XY$ of $\Gamma$ is tangent to $\omega$ and it passes through $N$ , compute $XY$ .

Geo Mock #9

by Bluesoul, Apr 1, 2025, 7:05 AM

Consider $\triangle{ABC}$ with $AB=12, AC=22$ . The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$ . Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$ . Extend $AT$ to $BC$ at $R$ , given $AR=16$ , find $[ABC]$ .

Geo Mock #6

by Bluesoul, Apr 1, 2025, 7:04 AM

Consider triangle $ABC$ with $AB=5, BC=8, AC=7$ , denote the incenter of the triangle as $I$ . Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$ , find the length of $QC$ .