Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
Geometry with orthocenter config
thdnder   6
N 2 hours ago by ohhh
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
6 replies
1 viewing
thdnder
Apr 29, 2025
ohhh
2 hours ago
Strange Inequality
anantmudgal09   40
N 2 hours ago by starchan
Source: INMO 2020 P4
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
Proposed by Kapil Pause
40 replies
anantmudgal09
Jan 19, 2020
starchan
2 hours ago
Finding Solutions
MathStudent2002   22
N 2 hours ago by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\]Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
22 replies
MathStudent2002
Jul 19, 2017
ihategeo_1969
2 hours ago
USAMO 2000 Problem 3
MithsApprentice   10
N 2 hours ago by HamstPan38825
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
10 replies
MithsApprentice
Oct 1, 2005
HamstPan38825
2 hours ago
Hard limits
Snoop76   7
N 2 hours ago by MihaiT
$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1, $ $b_{0}=1$, $ a_{n+1}=a_{n}+b_{n}$$ $ and $ $$ b_{n+1}=(2n+3)b_{n}+a_{n}$. Find $ \lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $ $ and $ $ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$
7 replies
Snoop76
Mar 25, 2025
MihaiT
2 hours ago
Additive combinatorics (re Cauchy-Davenport)
mavropnevma   3
N 2 hours ago by Orzify
Source: Romania TST 3 2010, Problem 4
Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements.

***
3 replies
mavropnevma
Aug 25, 2012
Orzify
2 hours ago
Ducks can play games now apparently
MortemEtInteritum   34
N 3 hours ago by HamstPan38825
Source: USA TST(ST) 2020 #1
Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks
picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors.
A move consists of an operation of one of the following three forms:

[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take
place, over all possible initial configurations.
34 replies
MortemEtInteritum
Nov 16, 2020
HamstPan38825
3 hours ago
Floor sequence
va2010   87
N 3 hours ago by Mathgloggers
Source: 2015 ISL N1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2}   \qquad  \textrm{and} \qquad    a_{k+1} = a_k\lfloor a_k \rfloor   \quad \textrm{for} \, k = 0, 1, 2, \cdots \]contains at least one integer term.
87 replies
va2010
Jul 7, 2016
Mathgloggers
3 hours ago
INMO 2019 P3
div5252   45
N 3 hours ago by anudeep
Let $m,n$ be distinct positive integers. Prove that
$$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$Further, determine when equality holds.
45 replies
div5252
Jan 20, 2019
anudeep
3 hours ago
My unsolved problem
ZeltaQN2008   1
N 3 hours ago by Adywastaken
Source: Belarus 2017
Find all funcition $f:(0,\infty)\rightarrow (0,\infty)$ such that for all any $x,y\in (0,\infty)$ :
$f(x+f(xy))=xf(1+f(y))$
1 reply
ZeltaQN2008
4 hours ago
Adywastaken
3 hours ago
a