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we can find one pair of a boy and a girl
orl   16
N an hour ago by ezpotd
Source: Vietnam TST 2001 for the 42th IMO, problem 3
Some club has 42 members. It’s known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.
16 replies
orl
Jun 26, 2005
ezpotd
an hour ago
J
H
teleporting wizard starts on point (2017, 101), 4 moves
parmenides51   1
N an hour ago by jasperE3
Source: 2018 USAIMEO #2 p5 (Mock AIME -USAJMO) https://artofproblemsolving.com/community/c594864h1572209p9658908
A teleporting wizard starts on the point $(2017, 101)$ and can teleport to other Cartesian coordinates with only $1$ of $4$ moves: $(x, y) \to (x + y, y)$, $(x, y) \to (x - y, y)$ when $x > y$, $(x, y) \to (x, x + y)$, and $(x, y) \to (x, y - x)$ when $y > x$.

(a) Let $P(x)$ be any polynomial with positive integer coefficients that passes through $(0, 0)$. Show that for all such $P(x)$, there exists a unique point on the curve where the wizard can land on.

(b) For each $P(x)$, let $S$ be this unique point. Find the equation of the graph that contains all potential $S$.
1 reply
parmenides51
Nov 17, 2023
jasperE3
an hour ago
J
H
IMO ShortList 2001, combinatorics problem 1
orl   33
N an hour ago by ihategeo_1969
Source: IMO ShortList 2001, combinatorics problem 1
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
33 replies
orl
Sep 30, 2004
ihategeo_1969
an hour ago
J
H
CooL geo
Pomegranat   0
2 hours ago
Source: Idk

In triangle \( ABC \), \( D \) is the midpoint of \( BC \). \( E \) is an arbitrary point on \( AC \). Let \( S \) be the intersection of \( AD \) and \( BE \). The line \( CS \) intersects with the circumcircle of \( ACD \), for the second time at \( K \). \( P \) is the circumcenter of triangle \( ABE \). Prove that \( PK \perp CK \).
0 replies
Pomegranat
2 hours ago
0 replies
J
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Coefficient Problem
P162008   2
N 2 hours ago by cazanova19921
Consider the polynomial $g(x) = \prod_{i=1}^{7} \left(1 + x^{i!} + x^{2i!} + x^{3i!} + \cdots + x^{(i-1)i!} + x^{ii!}\right)$
Find the coefficient of $x^{2025}$ in the expansion of $g(x).$
2 replies
P162008
Yesterday at 12:16 PM
cazanova19921
2 hours ago
J
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Shooting An Invisible Tank
Aryan27   0
2 hours ago
Source: 239 MO
An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $k$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the process is repeated. Find the smallest value of $k$ such that the cannon can definitely shoot the tank after some time.
0 replies
Aryan27
2 hours ago
0 replies
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Showing Tangency
Itoz   1
N 3 hours ago by ja.
Source: Own
The circumcenter of $\triangle ABC$ is $O$. Line $AO$ meets line $BC$ at point $D$, and there is a point $E$ on $\odot(ABC)$ such that $AE \perp BC$. Line $DE$ intersects $\odot(ABC)$ at point $F$. The perpendicular bisector of line segment $BC$ intersects line $AB$ at point $K$, and line $AB$ intersects $\odot(CFK)$ at point $L$.

Prove that $\odot(AFL)$ is tangent to $\odot (OBC)$.
1 reply
Itoz
Yesterday at 1:57 PM
ja.
3 hours ago
J
H
2-var inequality
sqing   7
N 3 hours ago by mathuz
Source: Own
Let $ a,b\geq 0 $. Prove that
$$ \frac{a }{a^2+2b^2+1}+ \frac{b }{b^2+2a^2+1}\leq \frac{1}{\sqrt{3}} $$$$ \frac{a }{2a^2+ b^2+2ab+1}+ \frac{b }{2b^2+ a^2+2ab+1} \leq \frac{1}{\sqrt{5}} $$$$ \frac{a }{2a^2+ b^2+ ab+1}+ \frac{b }{2b^2+ a^2+ ab+1} \leq \frac{1}{2} $$$$\frac{a }{a^2+2b^2+2ab+1}+ \frac{b }{b^2+2a^2+2ab+1}\leq \frac{1}{2} $$
7 replies
sqing
Today at 1:39 AM
mathuz
3 hours ago
J
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1988 USAMO Problem 4
ahaanomegas   31
N 3 hours ago by LeYohan
Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.
31 replies
ahaanomegas
Jul 27, 2011
LeYohan
3 hours ago
J
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How many numbers
brokendiamond   0
4 hours ago
How many 5-digit numbers can be formed using the digits 1, 3, 5, 7, 9 such that the smaller digits are not positioned between two larger digits?
0 replies
brokendiamond
4 hours ago
0 replies
J
a