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IMO 2008, Question 2
delegat   63
N 38 minutes ago by ezpotd
Source: IMO Shortlist 2008, A2
(a) Prove that
\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

(b) Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

Author: Walther Janous, Austria
63 replies
delegat
Jul 16, 2008
ezpotd
38 minutes ago
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Inequalities
sqing   13
N 5 hours ago by ytChen
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
13 replies
sqing
May 13, 2025
ytChen
5 hours ago
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System of Equations
P162008   1
N 5 hours ago by alexheinis
If $a,b$ and $c$ are real numbers such that

$(a + b)(b + c) = -1$

$(a - b)^2 + (a^2 - b^2)^2 = 85$

$(b - c)^2 + (b^2 - c^2)^2 = 75$

Compute $(a - c)^2 + (a^2 - c^2)^2.$
1 reply
P162008
Monday at 10:48 AM
alexheinis
5 hours ago
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Might be the first equation marathon
steven_zhang123   35
N Yesterday at 7:09 PM by lightsbug
As far as I know, it seems that no one on HSM has organized an equation marathon before. Click to reveal hidden textSo why not give it a try? Click to reveal hidden text Let's start one!
Some basic rules need to be clarified:
$\cdot$ If a problem has not been solved within $5$ days, then others are eligible to post a new probkem.
$\cdot$ Not only simple one-variable equations, but also systems of equations are allowed.
$\cdot$ The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
$\cdot$ Please indicate the domain of the solution to the equation (e.g., solve in $\mathbb{R}$, solve in $\mathbb{C}$).
Here's an simple yet fun problem, hope you enjoy it :P :
P1
35 replies
steven_zhang123
Jan 20, 2025
lightsbug
Yesterday at 7:09 PM
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Maximum number of empty squares
Ecrin_eren   0
Yesterday at 6:35 PM


There are 16 kangaroos on a giant 4×4 chessboard, with exactly one kangaroo on each square. In each round, every kangaroo jumps to a neighboring square (up, down, left, or right — but not diagonally). All kangaroos stay on the board. More than one kangaroo can occupy the same square. What is the maximum number of empty squares that can exist after 100 rounds?



0 replies
Ecrin_eren
Yesterday at 6:35 PM
0 replies
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THREE People Meet at the SAME. TIME.
LilKirb   7
N Yesterday at 5:33 PM by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
7 replies
LilKirb
Monday at 1:06 PM
hellohi321
Yesterday at 5:33 PM
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Quite straightforward
steven_zhang123   1
N Yesterday at 3:16 PM by Mathzeus1024
Given that the sequence $\left \{ a_{n} \right \} $ is an arithmetic sequence, $a_{1}=1$, $a_{2}+a_{3}+\dots+a_{10}=144$. Let the general term $b_{n}$ of the sequence $\left \{ b_{n} \right \}$ be $\log_{a}{(1+\frac{1}{a_{n}} )} ( a > 0 \text{and} a \ne 1)$, and let $S_{n}$ be the sum of the $n$ terms of the sequence $\left \{ b_{n} \right \}$. Compare the size of $S_{n}$ with $\frac{1}{3} \log_{a}{(1+\frac{1}{a_{n}} )} $.
1 reply
steven_zhang123
Jan 11, 2025
Mathzeus1024
Yesterday at 3:16 PM
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Inequalities
sqing   0
Yesterday at 2:23 PM
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
0 replies
sqing
Yesterday at 2:23 PM
0 replies
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Function and Quadratic equations help help help
Ocean_MathGod   1
N Yesterday at 11:26 AM by Mathzeus1024
Consider this parabola: y = x^2 + (2m + 1)x + m(m - 3) where m is constant and -1 ≤ m ≤ 4. A(-m-1, y1), B(m/2, y2), C(-m, y3) are three different points on the parabola. Now rotate the axis of symmetry of the parabola 90 degrees counterclockwise around the origin O to obtain line a. Draw a line from the vertex P of the parabola perpendicular to line a, meeting at point H.

1) express the vertex of the quadratic equation using an expression with m.
2) If, regardless of the value of m, the parabola and the line y=x−km (where k is a constant) have exactly one point of intersection, find the value of k.

3) (where I'm struggling the most) When 1 < PH ≤ 6, compare the values of y1, y2, and y3.
1 reply
Ocean_MathGod
Aug 26, 2024
Mathzeus1024
Yesterday at 11:26 AM
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System of Equations
P162008   1
N Yesterday at 10:30 AM by alexheinis
If $a,b$ and $c$ are complex numbers such that

$\frac{ab}{b + c} + \frac{bc}{c + a} + \frac{ca}{a + b} = -9$

$\frac{ab}{c + a} + \frac{bc}{a + b} + \frac{ca}{b + c} = 10$

Compute $\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c}.$
1 reply
P162008
Monday at 10:34 AM
alexheinis
Yesterday at 10:30 AM
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Inequalities
sqing   19
N Yesterday at 8:40 AM by sqing
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
19 replies
sqing
May 15, 2025
sqing
Yesterday at 8:40 AM
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Board problem with complex numbers
egxa   1
N Apr 18, 2025 by hectorraul
Source: All Russian 2025 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
1 reply
egxa
Apr 18, 2025
hectorraul
Apr 18, 2025
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Board problem with complex numbers
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Source: All Russian 2025 11.1
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egxa
211 posts
egxa
#1 h  Apr 18, 2025, 9:45 AM • 1 Y
Y by Tung-CHL
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
Z K Y
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hectorraul
363 posts
hectorraul
#2 h  Apr 18, 2025, 11:34 AM • 1 Y
Y by Quidditch
Consider the graph $G$ of $777$ nodes, one for each complex number and draw edges if the condition is held.

$\textbf{claim:}$ $G$ is form by exactly $17$ disconnected and simple path.
$\textbf{proof:}$ Condition is equivalent to $a-b = \pm i$ meaning that $a$ can only be connected to $a-i$ and $a+i$, this ensure that $G$ is a union of disconnected and simple path.
Suppose that we have $k$ of these path, each of length $l_1,l_2,...,l_k$. The we have that $\sum l_i = 760$ and $777 = \sum (l_i+1) = \sum l_i + k = 760+k$, then $k = 17$.$\square$

Now, by PHP one of these chains has at least $\lceil 777/17\rceil = 46$ nodes. So we have a complex number $c$ and $c+45i$ and its easy to check that
\[ c^2+(c+45i)^2+2025 = 2c(c+45i) \].
This post has been edited 4 times. Last edited by hectorraul, Apr 20, 2025, 6:25 PM
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