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Easy IMO 2023 NT
799786   137
N an hour ago by InftyByond
Source: IMO 2023 P1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
137 replies
799786
Jul 8, 2023
InftyByond
an hour ago
JBMO TST Bosnia and Herzegovina 2025 P3
Rotten_   7
N an hour ago by sqing
Let a, b, c be real numbers such that
\[
a + b + c = 0 \quad \text{and} \quad abc = -16.
\]Find the minimum value of the expression
\[
W = \frac{a^2 + b^2} {c} + \frac{b^2 + c^2} {a} + \frac{c^2 + a^2} {b}.
\]
7 replies
Rotten_
Yesterday at 9:24 AM
sqing
an hour ago
Prove two circles intersect on line BC
62861   76
N an hour ago by KevinYang2.71
Source: USA Winter TST for IMO 2019, Problem 1 and TST for EGMO 2019, Problem 2, by Merlijn Staps
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.

Merlijn Staps
76 replies
62861
Dec 10, 2018
KevinYang2.71
an hour ago
Interesting inequality
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b=2. $ Prove that
$$   \frac{1}{a^2}+\frac{1}{b^2}-\frac{k}{ab} \geq -\frac{(k+2)^2}{16} $$Where $ k\geq 6.$
$$   \frac{1}{a^2}+\frac{1}{b^2}-\frac{6}{ab} \geq -4$$$$  \frac{1}{a^2}+\frac{1}{b^2}-\frac{8}{ab}  \geq -\frac{25}{4} $$
5 replies
sqing
6 hours ago
sqing
an hour ago
Some Integers written in a black board
Kunihiko_Chikaya   9
N 2 hours ago by Namura
Source: 2009 Japan Mathematical Olympiad Finals, Problem 2
Let $ N$ be postive integer. Some integers are written in a black board and those satisfy the following conditions.

1. Any numbers written are integers which are from 1 to $ N$.

2. More than one integer which is from 1 to $ N$ is written.

3. The sum of numbers written is even.

If we mark $ X$ to some numbers written and mark $ Y$ to all remaining numbers, then prove that we can set the sum of numbers marked $ X$ are equal to that of numbers marked $ Y$.
9 replies
Kunihiko_Chikaya
Feb 21, 2009
Namura
2 hours ago
Inequality
JARP091   0
2 hours ago
Source: Own
$x, y, z \ge 0$. Prove that:
\[
\sum \frac{x^2}{51(x+y+z)^2 + 81x^2} \ge \frac{1}{180}
\]
0 replies
1 viewing
JARP091
2 hours ago
0 replies
A tournament
Omid Hatami   22
N 2 hours ago by de-Kirschbaum
Source: Iran TST 2008
Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.
22 replies
Omid Hatami
May 25, 2008
de-Kirschbaum
2 hours ago
Symmetric inequality with two directions
Tintarn   23
N 3 hours ago by Sakura-junlin
Source: Germany 2017, Problem 5
Prove that for all non-negative numbers $x,y,z$ satisfying $x+y+z=1$, one has
\[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]
23 replies
Tintarn
May 4, 2017
Sakura-junlin
3 hours ago
Guessing Point is Hard
MarkBcc168   33
N 3 hours ago by HamstPan38825
Source: IMO Shortlist 2023 G5
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.

Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.

Ivan Chan Kai Chin, Malaysia
33 replies
MarkBcc168
Jul 17, 2024
HamstPan38825
3 hours ago
The only 2D geometry on the test
v_Enhance   17
N 4 hours ago by KnowingAnt
Source: USA TSTST 2025/9
Let acute triangle $ABC$ have orthocenter $H$. Let $B_1$, $C_1$, $B_2$, and $C_2$ be collinear points which lie on lines $AB$, $AC$, $BH$, and $CH$, respectively. Let $\omega_B$ and $\omega_C$ be the circumcircles of triangles $BB_1B_2$ and $CC_1C_2$, respectively. Prove that the radical axis of $\omega_B$ and $\omega_C$ intersects the line through their centers on the nine-point circle of triangle $ABC$.

Ruben Carpenter
17 replies
v_Enhance
Jul 1, 2025
KnowingAnt
4 hours ago
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