Search found 312 matches
- Wed Jul 21, 2021 8:21 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 6
- Replies: 0
- Views: 47897
IMO 2021, Problem 6
Let $m \ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1,B_2, B_3, \ldots, B_m$ subsets of $A$. Assume that for each $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.
- Wed Jul 21, 2021 8:17 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 5
- Replies: 0
- Views: 47750
IMO 2021, Problem 5
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favorite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but ha...
- Wed Jul 21, 2021 8:15 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 4
- Replies: 0
- Views: 47245
IMO 2021, Problem 4
Let $\Gamma$ be a circle with center $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB, BC, CD$, and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond ...
- Wed Jul 21, 2021 8:12 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 3
- Replies: 0
- Views: 46995
IMO 2021, Problem 3
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX...
- Wed Jul 21, 2021 8:11 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 2
- Replies: 0
- Views: 47016
IMO 2021, Problem 2
Show that the inequality
\[\sum_{i=1}^{n} \sum_{j=1}^{n} (\sqrt{|x_{i} - x_{j}|}) \leq \sum_{i=1}^{n} \sum_{j=1}^{n} (\sqrt{|x_{i} +x_{j}|})\]
holds for all real numbers $x_1, \ldots, x_n.$
\[\sum_{i=1}^{n} \sum_{j=1}^{n} (\sqrt{|x_{i} - x_{j}|}) \leq \sum_{i=1}^{n} \sum_{j=1}^{n} (\sqrt{|x_{i} +x_{j}|})\]
holds for all real numbers $x_1, \ldots, x_n.$
- Wed Jul 21, 2021 8:07 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO 2021, Problem 1
- Replies: 2
- Views: 13358
IMO 2021, Problem 1
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
- Mon May 31, 2021 9:04 pm
- Forum: Introductions
- Topic: New here and introduction
- Replies: 1
- Views: 10566
- Wed Apr 21, 2021 7:24 pm
- Forum: Introductions
- Topic: New to forum
- Replies: 3
- Views: 18055
- Mon Apr 12, 2021 9:27 pm
- Forum: Social Lounge
- Topic: How to start my math journey?
- Replies: 2
- Views: 9368
Re: How to start my math journey?
Hi I'm Joy. Age of 20. But currently I'm very much interested in math problem solving. So, it would be very helpful if any one give me a list of book for learn the basic of approach for solving a mathematic problem. And also a list of basic topic which I need to learn first. Advance thanks to all o...
- Fri Apr 02, 2021 2:34 am
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 663868
Re: FE Marathon!
$\textbf{Problem 24}$ Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\] Source : IMOSL 1977 Actually this is IMO 1977/P6 . Hint: Try to prove that $f(n) \geq n$ $\forall n \in \mathbb{N}$. Then try to prove ...