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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Yesterday at 3:18 PM
0 replies
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Sequence of numbers in form of a^2+b^2
TheOverlord   12
N 8 minutes ago by ihategeo_1969
Source: Iran TST 2015, exam 1, day 1 problem 3
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
12 replies
TheOverlord
May 11, 2015
ihategeo_1969
8 minutes ago
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Perpendicular following tangent circles
buzzychaoz   20
N 23 minutes ago by ihategeo_1969
Source: China Team Selection Test 2016 Test 2 Day 2 Q6
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
20 replies
buzzychaoz
Mar 21, 2016
ihategeo_1969
23 minutes ago
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R to R FE
a_507_bc   10
N 23 minutes ago by jasperE3
Source: Baltic Way 2023/4
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$for reals $x, y$.
10 replies
a_507_bc
Nov 11, 2023
jasperE3
23 minutes ago
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Functional Equation from strings to a set of integers
Jia_Le_Kong   3
N 40 minutes ago by jasperE3
Source: own
Let $S$ be the set of all non-empty sub-strings of "Socrates".
Thus, "Soc", "crates" $\in S$ but "teS", "sorat", "setarcoS" $\notin S$.

Define the sum of two strings $x$ and $y$ as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be $x$. Then the sum of these two strings will be the string formed by all the letters between the first term of $x$ and the last term of $y$. For instance, "Socr" + "te" $=$ "Socrate" and "cra" + "ates" $=$ "crates".

Find all functions $f:S \rightarrow \{t : 0 \le t \le 49, \text{ } t \in \mathbb{Z}\}$ such that

$$f(x) + f(y) = 2f(x+y)$$
for all $x, y \in S$.

Bonus. What if we change "Socrates" to a string assumed to be of an arbitrary natural number length, with distinct letters (or symbols, if we need more)? The range $\{t : 0 \le t \le 49, \text{ } t \in \mathbb{Z}\}$ is still the same.
3 replies
1 viewing
Jia_Le_Kong
Jul 29, 2023
jasperE3
40 minutes ago
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Count the distinct values in 2025 fractions
Stuttgarden   1
N Mar 31, 2025 by RagvaloD
Source: Spain MO 2025 P1
Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]
1 reply
Stuttgarden
Mar 31, 2025
RagvaloD
Mar 31, 2025
J
Count the distinct values in 2025 fractions
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Source: Spain MO 2025 P1
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Stuttgarden
34 posts
Stuttgarden
#1 h  Mar 31, 2025, 1:04 PM
Y by
Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]
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RagvaloD
4894 posts
RagvaloD
#2 h  Mar 31, 2025, 3:50 PM • 1 Y
Y by internationalnick123456
Easy to prove, that $\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\dots,\left\lfloor\frac{2025}{44}\right\rfloor$ takes different values greater $45$ and $\left\lfloor\frac{2025}{45}\right\rfloor,\left\lfloor\frac{2025}{46}\right\rfloor,\dots,\left\lfloor\frac{2025}{2025}\right\rfloor$ take all values from $1$ to $45$ so in total $45+44=89$ different values
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