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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Product of consecutive terms divisible by a prime number
BR1F1SZ   0
3 minutes ago
Source: 2025 Francophone MO Seniors P4
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]is a multiple of $p$.
[/list]


0 replies
BR1F1SZ
3 minutes ago
0 replies
Fixed and variable points
BR1F1SZ   0
6 minutes ago
Source: 2025 Francophone MO Seniors P3
Let $\omega$ be a circle with center $O$. Let $B$ and $C$ be two fixed points on the circle $\omega$ and let $A$ be a variable point on $\omega$. We denote by $X$ the intersection point of lines $OB$ and $AC$, assuming $X \neq O$. Let $\gamma$ be the circumcircle of triangle $\triangle AOX$. Let $Y$ be the second intersection point of $\gamma$ with $\omega$. The tangent to $\gamma$ at $Y$ intersects $\omega$ at $I$. The line $OI$ intersects $\omega$ at $J$. The perpendicular bisector of segment $OY$ intersects line $YI$ at $T$, and line $AJ$ intersects $\gamma$ at $P$. We denote by $Z$ the second intersection point of the circumcircle of triangle $\triangle PYT$ with $\omega$. Prove that, as point $A$ varies, points $Y$ and $Z$ remain fixed.
0 replies
BR1F1SZ
6 minutes ago
0 replies
Use 3d paper
YaoAOPS   7
N 11 minutes ago by EGMO
Source: 2025 CTST p4
Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?
7 replies
YaoAOPS
Mar 6, 2025
EGMO
11 minutes ago
Cyclic ine
m4thbl3nd3r   2
N 30 minutes ago by m4thbl3nd3r
Let $a,b,c>0$ such that $a^2+b^2+c^2=3$. Prove that $$\sum \frac{a^2}{b}+abc \ge 4$$
2 replies
m4thbl3nd3r
Yesterday at 3:34 PM
m4thbl3nd3r
30 minutes ago
No more topics!
Sequence with non-positive terms
socrates   8
N Apr 12, 2025 by ray66
Source: Baltic Way 2014, Problem 2
Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and \[a_{i+1} - 2a_i + a_{i-1} = a^2_i\] for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$
8 replies
socrates
Nov 11, 2014
ray66
Apr 12, 2025
Sequence with non-positive terms
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G H BBookmark kLocked kLocked NReply
Source: Baltic Way 2014, Problem 2
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socrates
2105 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and \[a_{i+1} - 2a_i + a_{i-1} = a^2_i\] for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$
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mavropnevma
15142 posts
#2 • 3 Y
Y by FlakeLCR, Adventure10, Mango247
The condition $a_{i+1} - 2a_i + a_{i-1} = a^2_i$ for $i = 1, 2, \ldots, N - 1$ is a herring; it is enough to know $a_{i+1} - 2a_i + a_{i-1} \geq 0$.

Consider $x_i = \textrm{e}^{-a_i}$; then $x_1^2 = \textrm{e}^{-2a_i} \geq \textrm{e}^{-a_{i-1} - a_{i+1}} = x_{i-1}x_{i+1}$, i.e. the sequence $(x_i)_{i=0}^{i=N}$ is log-concave, and as such, it is known (and easy to prove) it is unimodal, i.e. for some $0\leq j \leq N$ we have $x_0\leq x_1\leq \cdots \leq x_{j-1} \leq x_j \geq x_{j+1} \geq \cdots \geq x_N$.
But that means $0 = a_0 \geq a_1\geq \cdots \geq a_{j-1} \geq a_j \leq a_{j+1} \leq \cdots \leq a_N = 0$.
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Tintarn
9042 posts
#3 • 3 Y
Y by Kameawtamprooz, Adventure10, Mango247
mavropnevma wrote:
Consider $x_i = \textrm{e}^{-a_i}; \dotsc$
Why so complicated?
Just assume the maximum of the sequence would be at some $i$ not equal to $0$ or $N$.
Then $a_i \ge a_{i+1}$ and $a_i \ge a_{i-1}$ which gives $0 \ge a_{i-1}+a_{i+1}-2a_i=a_i^2$ and thus $a_i=0$ which gives in every case the result that the maximum of the sequence is $0$.
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mavropnevma
15142 posts
#4 • 4 Y
Y by Tintarn, FlakeLCR, Adventure10, Mango247
Tintarn wrote:
Why so complicated?
Just because I wanted to bring into picture the wide-spread literature on log-convex and log-concave sequences; of course, the whole thing is trivial.
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MADharm
96 posts
#5 • 2 Y
Y by Adventure10, Mango247
$a^2_i+a^2_{i-1}+a^2_{i-2}+...+a^2_2+a^2_1 = -(a_i+a_1)$

LHS is nonnegative, then RHS must nonnegative too. We are done (cmiiw)
This post has been edited 2 times. Last edited by MADharm, Dec 6, 2018, 9:36 AM
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Williamgolly
3760 posts
#6
Y by
Suppose $a_i >0.$ Then, $2a_i+a_i^2=a_{i-1}+a_{i+1} >0.$ Then, either $a_{i-1}$ or $a_{i+1}$ is less than $0.$ We can continue using this process until we get $a_0$ or $a_N$ equal to $0,$ which is contradiction. Therefore, $a_i<0$ for all $1 \leq i \leq N-1.$
This post has been edited 2 times. Last edited by Williamgolly, Nov 13, 2020, 12:28 AM
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rafaello
1079 posts
#7
Y by
Claim 1. Suppose $a_{i+1}> 0$, $a_{i+1}-a_i>0$ for some $n-2\geq i\geq 0$. Then, $a_{i+2}> 0$ and $a_{i+2}-a_{i+1}>0$.
Proof.
We have $$a_{i+2}=a_{i+1}^2+2a_{i+1}-a_{i}> 0$$and $$a_{i+2}-a_{i+1}=a_{i+1}^2+a_{i+1}-a_{i}> 0.$$

Claim 2. Suppose $a_{i-1}\leq 0$ for some $n-1\geq i\geq 1$. Then, $a_i\leq 0$.
Proof.
Suppose otherwise, i.e. $a_i>0$ and $a_{i-1}\leq 0$, hence
$$a_{i+1}=a_i^2+2a_i-a_{i-1}> 0$$and $a_{i+1}-a_i=a_i^2+a_i-a_{i-1}>0$. By the first claim we have that $a_n>0$, which is a contradiction.

Now by the second claim, we are done.
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laikhanhhoang_3011
637 posts
#8
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s
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ray66
35 posts
#9
Y by
Because $a_{i+1}+a_{i-1} \ge 2a_i$, we either have some increasing or decreasing sequence. But because $a_0$ and $a_n$ are both 0, the entire sequence has to be 0.
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