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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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0 replies
jlacosta
Apr 2, 2025
0 replies
Too Bad I'm Lactose Intolerant
hwl0304   218
N 12 minutes ago by Maximilian113
Source: 2018 USAMO Problem 1/USAJMO Problem 2
Let \(a,b,c\) be positive real numbers such that \(a+b+c=4\sqrt[3]{abc}\). Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]
218 replies
+2 w
hwl0304
Apr 18, 2018
Maximilian113
12 minutes ago
Please support!
warriorsin7   0
an hour ago
warriorsin7
an hour ago
0 replies
SMT Online 2025 Certificates/Question Paper/Grading
techb   1
N 3 hours ago by Inaaya
It is May 1st. I have been anticipating the arrival of my results displayed in the awards ceremony in the form of a digital certificate. I have unfortunately not received anything. I have heard from other sources(AoPS, and the internet), that the certificates generally arrive at the end of the month. I would like to ask the organizers, or the coordinators of the tournament, to at least give us an ETA. I would like to further elaborate on the expedition of the release of the Question Papers and the grading. The question papers would be very helpful to the people who have taken the contest, and also to other people who would like to solve them. It would also help, as people can discuss the problems that were given in the test, and know different strategies to solve a problem they have solved. In regards to the grading, it would be a crucial piece of evidence to dispute the score shown in the awards ceremony, in case the contestant is not satisfied.
1 reply
techb
3 hours ago
Inaaya
3 hours ago
9 Did I make the right choice?
Martin2001   27
N 3 hours ago by ninjaforce
If you were in 8th grade, would you rather go to MOP or mc nats? I chose to study the former more and got in so was wondering if that was valid given that I'll never make mc nats.
27 replies
Martin2001
Apr 29, 2025
ninjaforce
3 hours ago
No more topics!
Sort of additive function
tenniskidperson3   112
N Apr 27, 2025 by anudeep
Source: 2015 USAJMO problem 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)
112 replies
tenniskidperson3
Apr 29, 2015
anudeep
Apr 27, 2025
Sort of additive function
G H J
Source: 2015 USAJMO problem 4
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Arrowhead575
2281 posts
#102
Y by
Is that equation citable? I simplified it down to Cauchy's
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jasperE3
11281 posts
#103
Y by
Evan Chen A.3 says
Quote:
In general it is usually okay to cite a result that is (i) named, and (ii) does not trivialize
the given problem.
so it should be fine.
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Eagle_Student5
5 posts
#104 • 2 Y
Y by centslordm, megarnie
Solved with HamstPan38825.

We claim the only solutions are $f(x)=kx+c$, and it is easy to see that this works.

Now, define $x=a$, $y=a+d$, $z=a+2d$, and $t=a+3d$. Putting these into the equation gives us
\[f(a)+f(a+3d)=f(a+d)+f(a+2d)\]If we subtract $d$ from everything, this simplifies to
\[f(a-d)+f(a+2d)=f(a)+f(a+d)\]Then, if we add the two equations together and cancel, we get
\[f(a-d)+f(a+3d)=2f(a+d)\]This is Jensen's Functional Inequality, which we know has solution $f(x)=kx+c$, but only if $d\neq0$. So if $d=0$, then $a=x=y=z=t$, but this contradicts the fact that $x<y<z<t$, so $d$ can't be $0$, so we're done. $\square$
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Ritwin
156 posts
#105 • 1 Y
Y by LLL2019
Wrong

:oops_sign: my solution was wrong :blush:
This post has been edited 1 time. Last edited by Ritwin, May 13, 2022, 3:02 AM
Reason: spoilered the wrong solution :blush:
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megarnie
5603 posts
#106
Y by
Rewritten solution:

Let $P(x,y,z,t)$ denote the given assertion and $d$ be some nonzero rational number.

$P(x,x+d,x+2d,x+3d): f(x)+f(x+3d)=f(x+d)+f(x+2d)$.

$P(x-d,x,x+d,x+2d): f(x-d)+f(x+2d)=f(x)+f(x+d)$.

Adding the equations gives $f(x-d)+f(x)+f(x+2d)+f(x+3d)=2f(x+d)+f(x)+f(x+2d)\implies f(x-d)+f(x+3d)=2f(x+d)$.

Since $x$ and $d$ can be any rational numbers, this simplifies to $f(x)+f(y)=2f\left(\frac{x+y}{2}\right)$. This is Jensen's FE, but I will solve it here.

Let $g(x)=f(x)-f(0)$. Then $g(x)+g(y)+2f(0)=2g\left(\frac{x+y}{2}\right)+2f(0)\implies g(x)+g(y)=2g\left(\frac{x+y}{2}\right)$ with the additional constraint that $g(0)=0$. Let $Q(x,y)$ be this new assertion.

$Q(x,0): g(x)=2g\left(\frac{x}{2}\right)\implies g\left(\frac{x}{2}\right)=\frac{g(x)}{2}$.

Thus, $g(x)+g(y)=g(x+y)$, so $g$ is Cauchy, which is well-known to have solutions $g(x)=kx$ for some constant $k$ over the rationals.

So $f(x)=kx+f(0)$, $\boxed{f(x)=kx+c}$ for rational constants $k$ and $c$. To show that this works, we have $kx+c+kt+c=ky+c+kz+c\iff kx+kt=ky+kz$. Note if $k=0$, then constant solution obviously works. If not, then $kx+kt=ky+kz\iff x+t=y+z$, which is true since they form an arithmetic progression.
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ZETA_in_olympiad
2211 posts
#107
Y by
Answer
Proof
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brainfertilzer
1831 posts
#108
Y by
The only solutions are $\boxed{f(x) = ax + b}$ for all $x$, where $a,b\in\mathbb{Q}$. These work since
\[ (ax + b) + (at + b) = (ay + b) + (az + b)\iff a(x + t - y -z) = 0,\]which is true since $x + t = y + z$. It remains to show that these solutions are the only ones.

Take $\{x,y,z,t\} = \{a, a + d/2, a + d, a + 3d/2\}$ and $\{x,y,z,t\} = \{a + d/2, a + d, a + 3d/2, a + 3d\}$ separately to get the equations
\[ f(a) + f(a + 3d/2) = f(a + d/2) + f(a + d)\qquad (1),\]\[ f(a + d) + f( a + 3d/2) = f(a +d/2) + f(2d)\qquad (2).\]Subtracting $(1)$ from $(2)$ gives $f(a + d) - f(a) = f(a + 2d) - f(a + d)\implies f(a + d) = \frac{f(a) + f(a + 2d)}{2}$. Using the substitution $(x,y) = (a, a+ 2d)$, we have $f\left(\frac{x + y}{2}\right) = \frac{f(x)+ f(y)}{2}$ for all $x,y\in\mathbb{Q}$. Hence, $f$ satisfies Jensen's FE over the rationals, so $f$ is linear.
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rayfish
1121 posts
#109
Y by
The only solutions are linear $f$, or $f(x)=mx+n$ for rational $m$ and $n$, which works. Now, we show these are the only solutions.

Let rational numbers $a<b<c<d<e$ be in arithmetic progression. Then
\begin{align*}
f(b) + f(e) &= f(c) + f(d)\\
f(b) + f(c) &= f(a) + f(d)\\
\end{align*}Subtracting, we get $f(e)-f(c) = f(c)-f(a)$, so $f$ is linear over all arithmetic progressions in $\mathbb{Q}$. In particular, $f$ is linear for all integer inputs. Let $g$ be a linear function over $\mathbb{Q}$ such that $g(x)=f(x)$ for all integral $x$. We will show that $g=f$.

Let $r=\frac p q$ be a nonzero rational number, where $p$ and $q$ are nonzero integers. Since $f$ is linear over $\{ kr \mid k\in\mathbb{Z} \}$, and $f(0)=g(0)$ and $f(p) = g(p)$, we know that $f=g$ over that set, so $f(r)=g(r)$. Hence, $f=g$, as desired.
This post has been edited 5 times. Last edited by rayfish, Dec 5, 2022, 2:45 AM
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RedFireTruck
4221 posts
#110 • 1 Y
Y by BluePoliceCar
Let the average be $a$ and $d=z-a$. Then, $f(a-3d)+f(a+3d)=f(a-d)+f(a+d)$. We can plug in $a+2d$ for $a$ and we get $f(a+5d)-f(a+3d)=f(a+d)-f(a-d)$. We add these equation to get $f(a+5d)+f(a-3d)=2f(a+d)$. We can plug in $a-d$ for $a$ to get $f(a+4d)+f(a-4d)=2f(a)$. We can plug in $4d$ for $d$ to get $f(a+d)+f(a-d)=2f(a)$ for all rationals $a$ and $d$. It is easy to see by induction that $f(cx)=cf(x)-(c-1)f(0)$ for integers $c\ge 1$. This means that $f(p)=qf(\frac{p}{q})-(q-1)f(0)$ so $f(\frac{p}{q})=\frac{f(p)+(q-1)f(0)}{q}=\frac{pf(1)+(q-p)f(0)}{q}=\frac{p}{q}f(1)+(1-\frac{p}{q})f(0)=\frac{p}{q}(f(1)-f(0))+f(0)$ so $f(x)=x(f(1)-f(0))+f(0)=mx+b$ for all rational $x$. Plugging this back in, all $f(x)=mx+b$ work.
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asdf334
7585 posts
#111 • 1 Y
Y by BluePoliceCar
WLOG assume that $f(0)=0$. We may now consider the quantities $f(kd)$ for some integer $k$ and positive rational $d$; by induction we find that the values $f(2kd)$ actually form an arithmetic sequence. This immediately solves the problem over $\mathbb{Z}$ and subsequently $\mathbb{Q}$. $\blacksquare$

(This is essentially the same as @2above.)

Motivation
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HamstPan38825
8857 posts
#112
Y by
Notice that for all $a$ and $d>0$,
\begin{align*}
f(a+d) + f(a+2d) &= f(a) + f(a+3d) \\
f(a) + f(a+d) &= f(a-d) + f(a+2d).
\end{align*}Thus we have $$f(a+3d) + f(a-d) = 2f(a+d),$$so $f$ satisfies Jensen's functional equation. As a result, $f(x) = x+c$ for constants $c$ are the only solutions.
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Infinity_Integral
306 posts
#113
Y by
We claim only all $f(x)=mx+c, m,c\in\mathbb{Q}$ work.
Note that these functions work by checking.
For any $a,d$ we have
$$f(a+d)+f(a+4d)=f(a+2d)+f(a+3d)$$$$f(a+2d)+f(a+5d)=f(a+3d)+f(a+4d)$$So we get $f(a+d)+f(a+5d)=2f(a+3d)$, equivalent to
$$f(x-y)+f(x+y)=2f(x)$$Doing induction similar to the Cauchy Functional Equation over $\mathbb{Q}$ gives the solution set.
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OlympusHero
17020 posts
#114
Y by
Same solution as many others, but I'll post here anyway.

writeup
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NamelyOrange
502 posts
#115
Y by
Question: would Jensen's functional equation work here?
This post has been edited 1 time. Last edited by NamelyOrange, Oct 14, 2023, 4:22 PM
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anudeep
159 posts
#116
Y by
We claim $f$ is of the form $\alpha x+\beta$ for some $\alpha, \beta\in\mathbb{Q}$, easy to check why they work and we shall show that these are the only solutions to $f$.
Consider five terms of arithmetic progression namely $a,a+d, \ldots, a+4d$, with $a,d\in\mathbb{Q}$ and $d>0$. Plug these terms in the following manner,
\begin{align*}
f(a+d)-f(a)&=f(a+3d)-f(a+a+2d)\\
f(a+2d)-f(a+d)&=f(a+4d)-f(a+3d).
\end{align*}Adding two of the above equation we deduce to the following,
$$f(a+2d)=\cfrac{f(a+4d)+f(a)}{2}.$$Aha! this resembles the ubiquitous Jensen's equation and turns out it is precisely that as one may substitute $u=a$ and $v=a+4d$ ( this is because $(u,v)$ can generate the whole of $\mathbb{Q}\times\mathbb{Q}$). $\square$
This post has been edited 2 times. Last edited by anudeep, Apr 27, 2025, 2:44 PM
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