Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!

G
Topic
First Poster
Last Poster
MATHirang MATHibay 2025 (Orals, 1.3.2)
BinariouslyRandom   0
36 minutes ago
Find the third smallest number \( n \) which satisfies all the following properties:
\(\begin{array}{ll}
\bullet & n \text{ is a prime number}, \\
\bullet & n \text{ has more than 3 digits when written in base } 3, \\
\bullet & n \text{ has more than one unique digit in base } 3, \\
\bullet & n \text{ is a palindrome in base } 3.
\end{array}\)
Give your answer in base 10.
0 replies
1 viewing
BinariouslyRandom
36 minutes ago
0 replies
Word problem of Absolute-Value Function
Sukardy   1
N 42 minutes ago by Mathzeus1024
Please help me solve the following word problems. It is about absolute-value function. Any sketch of function's graph will be greatly appreciated.

Problem:
A combat plane is flying $50$ m above the sea. The plane will aim an enemy submarine in the distance of $100$ m. To be aimed on the target, a missile shot by the plane plunges into the sea with a depth of $30$ m. Suppose the surface of the sea to be $X$-axis and the plane's motion is represented by a function $f(x) = p|x|+q$ for $p, q \in \mathbb{R}$. Find the value of $p + q$.
1 reply
Sukardy
Nov 6, 2019
Mathzeus1024
42 minutes ago
Functional Equations
burntpizza001   1
N 2 hours ago by alexheinis
Let $f:R\rightarrow{}R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f(x)$, $\forall x\in R$.

(i) If $f(0)  = 5$ then minimum possible number of values of $x$ satisfying $f(x)=5$, for $x\in[0,170]$ is?

(ii) Graph of $y=f(x)$$ is symmetrical about which line?
1 reply
burntpizza001
May 25, 2025
alexheinis
2 hours ago
Binomial Sum
P162008   1
N 2 hours ago by alexheinis
If $f(x,n) = (x + n)^n - \binom{n}{1} (x + n - 1)^n + \binom{n}{2} (x + n - 2)^n - \cdots (-1)^n x^n$ then

A) $\sum_{r=1}^{4} f(4,r) = 33$

B) $\sum_{r=1}^{4} f(r,4) = 96$

C) $\sum_{r=1}^{5} f(5,r) = 129$

D) $\sum_{r=1}^{5} f(r,5) = 258$
1 reply
P162008
May 26, 2025
alexheinis
2 hours ago
AP calc?
Thayaden   31
N 3 hours ago by Mathzeus1024
How are we all feeling on AP calc guys?
31 replies
Thayaden
May 20, 2025
Mathzeus1024
3 hours ago
Inequalities
sqing   3
N 4 hours ago by sqing
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$
3 replies
sqing
Yesterday at 2:47 AM
sqing
4 hours ago
How can i prove this?
marxs01   1
N 5 hours ago by alexheinis
If $\frac{\cos{x}}{\cos{y}} + \frac{\sin{x}}{\sin{y}} = -1$ prove that $\frac{\cos^3{y}}{\cos{x}} + \frac{\sin^3{y}}{\sin{x}} = 1$
1 reply
marxs01
Yesterday at 10:56 PM
alexheinis
5 hours ago
Inequalities
sqing   1
N Today at 3:01 AM by sqing
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq  43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq  65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq  91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq  122$$
1 reply
sqing
Yesterday at 2:31 PM
sqing
Today at 3:01 AM
Can anyone help me with this inequality problem
a9opsow_   3
N Today at 1:38 AM by sqing
Let a, b, c be nonnegative real numbers, not all equal. Prove that

\frac{(a - bc)^2 + (b - ca)^2 + (c - ab)^2}{(a - b)^2 + (b - c)^2 + (c - a)^2} \geq \frac{1}{2}.
3 replies
a9opsow_
Yesterday at 11:50 AM
sqing
Today at 1:38 AM
Inequalities
sqing   21
N Today at 12:37 AM by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc +  abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc  + \frac{2}{5}abc  \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc  + \frac{7}{20}abc  \leq \frac{2381411}{26460}$$
21 replies
sqing
May 21, 2025
sqing
Today at 12:37 AM
a