Prove that for each real number $r > 2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x\rfloor$.
Note: $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
Search found 264 matches
- Wed Jun 09, 2021 4:56 pm
- Forum: Asian Pacific Math Olympiad (APMO)
- Topic: APMO 2021 P1
- Replies: 2
- Views: 8476
- Mon Jun 07, 2021 1:39 pm
- Forum: Junior Level
- Topic: Draw a Circle
- Replies: 2
- Views: 10503
Re: Draw a Circle
There are only finitely many lines that goes through at least $2$ of the points. Let's choose a line $l$ not parallel to any of them. So, $l$ goes through at most one point in that plane. Now translate the line $l$ such that it goes through a point in that plane and the number of points on both side...
- Sun Jun 06, 2021 3:59 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31886
Problem 09
Let $n\geq3$ be a fixed positive integer. A function $f:\mathbb{R}^2\to\mathbb{R}$ has the property that for any points $P_1, P_2,\cdots,P_n$ which are vertices of a regular $n$-gon, we have \[\sum_{i=1}^{n}f(P_i)=0\]
Prove that $f(P) = 0$ for all points $P\in\mathbb{R}^2$.
Prove that $f(P) = 0$ for all points $P\in\mathbb{R}^2$.
- Sun Jun 06, 2021 3:33 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31886
Re: Problem 8
There is a game show with $100$ contestants labelled $1, 2, \ldots, 100$. There are $100$ doors in this game show, each having a distinct hidden number from the set $\{1, \ldots, 100\}$. The contestants are not allowed to communicate with each other during the game. During the game, each contestant...
- Fri Jun 04, 2021 7:48 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31886
Problem 7
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that \[\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\geq\frac32\]
- Fri Jun 04, 2021 7:08 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31886
Solution to Problem 06
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 $AB$ at $B$. Prove th...
- Fri Jun 04, 2021 1:45 am
- Forum: Algebra
- Topic: Inequality Marathon
- Replies: 9
- Views: 11040
Problem 03
Let $a_1,a_2,a_3,\cdots,a_n$ be positive real numbers where $n\geq2, n\in\mathbb{N}$.
Let $s=a_1+a_2+a_3+\cdots+a_n$.
Prove that \[\frac{a_1}{s-a_1}+\frac{a_2}{s-a_2}+\frac{a_3}{s-a_3}+\cdots+\frac{a_n}{s-a_n}\geq\frac{n}{n-1}\]
Let $s=a_1+a_2+a_3+\cdots+a_n$.
Prove that \[\frac{a_1}{s-a_1}+\frac{a_2}{s-a_2}+\frac{a_3}{s-a_3}+\cdots+\frac{a_n}{s-a_n}\geq\frac{n}{n-1}\]
- Fri Jun 04, 2021 12:44 am
- Forum: Algebra
- Topic: Inequality Marathon
- Replies: 9
- Views: 11040
Re: Problem 02
Reposting an old problem... Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=5$ Find the minimum value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+16}+\sqrt{d^2+25}$ Solution : $\text{Lemma :}$ \[\sqrt{a_1^2+b_1^2}+\sqrt{a_2^2+b_2^2}\geq\sqrt{\left(a_1+a_2\right)^2+\left(b_1+b_2\right)^2}\] Fo...
- Wed Jun 02, 2021 11:44 pm
- Forum: College / University Level
- Topic: Maybe not as hard as you see
- Replies: 8
- Views: 21548
Re: Maybe not as hard as you see
Let's imagine a big rectangle $ABCD$ (named in clockwise direction) inside of which a closed loop is lying. Now start moving the side $AB$ towards $CD$ until it touches the closed loop. Do this for every sides, pull them towards the opposite side until they touches the loop. Following this algorith...
- Wed Jun 02, 2021 1:02 pm
- Forum: National Math Camp
- Topic: Special Problem Marathon
- Replies: 38
- Views: 31886
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...