Search found 5 matches
- Wed Jun 02, 2021 2:27 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34263
Re: Special Problem Marathon
$\textbf{Problem 6:}$ Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to ...
- Wed Jun 02, 2021 1:48 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34263
Re: Special Problem Marathon
Edited [I hope now it's okay]
- Wed Jun 02, 2021 12:57 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34263
Re: Special Problem Marathon
Problem 4: Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $$f(x + y) + f(x)^2f(y) = f(y)^3 + f(x + y)f(x)^2$$ for all $x, y \in \mathbb{Z}$. $\textbf{Solution (P4)}$ Here, the equation of the question is, $$f(x+y) + f(x)^2f(y) = f(y)^3+f(x+y)f(x)^2$$ Now, for $x=0,y=a$, we get...
- Tue Jun 01, 2021 10:26 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34263
Re: Special Problem Marathon
$\textbf{Problem 3:}$
Let $a, b, c, d$ be positive real numbers such that $a + b + c + d = 1$. Prove that
,
$$6(a^3 + b^3 + c^3 + d^3) \geq (a^2 + b^2 + c^2 + d^2) + \dfrac{1}{8}$$
Let $a, b, c, d$ be positive real numbers such that $a + b + c + d = 1$. Prove that
,
$$6(a^3 + b^3 + c^3 + d^3) \geq (a^2 + b^2 + c^2 + d^2) + \dfrac{1}{8}$$
- Tue Jun 01, 2021 10:13 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 34263
Re: Special Problem Marathon
$\textbf{Solution (P2):}$ Let $D_i$ and $ M_i$ be the number of ball of color $C_i$ in box $B_i$ and not in box $B_i$ respectively. And $S_i$ be the number of ball in $B_i$. For the sake of contradiction , lets assume that, $$D_i > \dfrac{1}{2} S_i \ \forall \ i \in \{1,2,3, \dots, N\}$$ $$\Rightarr...