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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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My Unsolved Problem
ZeltaQN2008   1
N a minute ago by ZeltaQN2008
Source: IDK
Let \( P(x) = x^{2024} + a_{2023}x^{2023} + \cdots + a_1x + a_0 \) be a polynomial with real coefficients.

(a) Suppose that \( 2023a_{2023}^2 - 4048a_{2022} < 0 \). Prove that the polynomial \( P(x) \) cannot have 2024 real roots.

(b) Suppose that \( a_0 = 1 \) and \( 2023(a_1^2 + a_2^2 + \cdots + a_{2023}^2) \leq 4 \). Prove that \( P(x) \geq 0 \) for all real numbers \( x \).
1 reply
ZeltaQN2008
May 29, 2025
ZeltaQN2008
a minute ago
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Pairs of integers that satisfy divisibility
ACGNmath   1
N 16 minutes ago by ACGNmath
Find all pairs $(m,n)$ of positive integers such that $n^2-2n\mid m^2+2m$ and $n^2-m\mid m^2+n$.
1 reply
ACGNmath
Jun 3, 2025
ACGNmath
16 minutes ago
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inequality
SunnyEvan   4
N 24 minutes ago by sqing
Let $ x,y \geq 0 ,$ such that : $ \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $$ x^2+y^2-xy \leq x+y $$$$ (x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x+1)^2+(y+1)^2 \leq 5 $$$$ (x+2)^2+(y+2)^2 \leq 13 $$
4 replies
SunnyEvan
2 hours ago
sqing
24 minutes ago
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You would not believe your eyes...
willwin4sure   12
N 33 minutes ago by pi271828
Source: USA TSTST 2020 Problem 9, by Nikolai Beluhov
Ten million fireflies are glowing in $\mathbb{R}^3$ at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may ever occupy the same point.

Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies.

Nikolai Beluhov
12 replies
willwin4sure
Jan 25, 2021
pi271828
33 minutes ago
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Ratio of incircle and excircle ? - JBMO Shortlist
WakeUp   1
N Oct 30, 2010 by Luis González
A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.
1 reply
WakeUp
Oct 30, 2010
Luis González
Oct 30, 2010
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Ratio of incircle and excircle ? - JBMO Shortlist
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Reveal topic tags
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geometry proposed
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WakeUp
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WakeUp
#1 h  Oct 30, 2010, 8:19 PM • 2 Y
Y by Adventure10, Mango247
A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.
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Luis González
4151 posts
Luis González
#2 h  Oct 30, 2010, 10:25 PM • 1 Y
Y by Adventure10
$(I,r),$ $(I_a,r_a),$ $(I_b,r_b),$ $(I_c,r_c)$ denote the incircle and three excircles against $A,B,C.$ Let $(U_a,\rho_a),$ $(U_b,\rho_b),$ $(U_b,\rho_c)$ be the mixtilinear incircles against $A,B,C$ and $(V_a, \varrho_a),$ $(V_b,\varrho_b),$ $(V_c,\varrho_c)$ their corresponding mixtilinear excircles. $(U_a,\rho_a)$ and $(V_a, \varrho_a)$ are tangent to rays $AB,AC$ through $M,N$ and $X,Y$ respectively. It's well-known that lines $MN$ and $XY$ pass through the incenter $I$ and A-excenter $I_a,$ respectively. Thus, from the similar kites $AMU_aN \sim AXV_aY,$ it follows that

$\alpha=\frac{\rho_a}{\varrho_a}=\frac{AI}{AI_a}=\frac{r}{r_a}$

Similarly, we have the ratios: $ \beta= \frac{\rho_b}{\varrho_b}=\frac{r}{r_b} \ , \ \gamma=\frac{\rho_c}{\varrho_c}=\frac{r}{r_c}$

$\Longrightarrow \alpha+\beta+\gamma=\frac{\rho_a}{\varrho_a}+\frac{\rho_b}{\varrho_b}+\frac{\rho_c}{\varrho_c}=r \left( \frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c} \right)=r \cdot \frac{1}{r}=1.$
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