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jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Geometry
Lukariman   5
N 7 minutes ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
5 replies
Lukariman
Yesterday at 12:43 PM
Lukariman
7 minutes ago
inq , not two of them =0
win14   0
13 minutes ago
Let a,b,c be non negative real numbers such that no two of them are simultaneously equal to 0
$$\frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a} \ge \frac{5}{2\sqrt{ab + bc + ca}}.$$
0 replies
win14
13 minutes ago
0 replies
IMO Genre Predictions
ohiorizzler1434   62
N 18 minutes ago by ehuseyinyigit
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
62 replies
ohiorizzler1434
May 3, 2025
ehuseyinyigit
18 minutes ago
Iranians playing with cards module a prime number.
Ryan-asadi   1
N 26 minutes ago by ItzsleepyXD
Source: Iranian Team Selection Test - P2
Let $p$ be an arbitrary prime number.we have a deck of cards which a number is written on the back of each of them. such that for every $i \in \{1,…,p-1\}$ we have at most one card with number $i$ written on its back and we also have exactly one card with number zero on its back.we want to design a game in which we need to decide for every two cards $X,Y$ witch one wins from the other one using the following rules.

$(I)$: If $x$ wins from $y$ and also $y$ wins from $z$ , then $x$ wins from $z$.
$(II)$: If $x$ doesn’t fail $y$ and $z$ doesn’t fail $t$ , then in case of existing of both cards $y+t$ and $x+z$ module $p$, card $x+z$ also doesn’t fail $y+t$.

What is the maximum number of cards which designing such game is possible?

1 reply
Ryan-asadi
an hour ago
ItzsleepyXD
26 minutes ago
No more topics!
Unexpected FE
Taco12   17
N Apr 25, 2025 by Ilikeminecraft
Source: 2023 Fall TJ Proof TST, Problem 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $x$ and $y$, \[ f(2x+f(y))+f(f(2x))=y. \]
Calvin Wang and Zani Xu
17 replies
Taco12
Oct 6, 2023
Ilikeminecraft
Apr 25, 2025
Unexpected FE
G H J
Source: 2023 Fall TJ Proof TST, Problem 3
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Taco12
1757 posts
#1 • 2 Y
Y by ItsBesi, rightways
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $x$ and $y$, \[ f(2x+f(y))+f(f(2x))=y. \]
Calvin Wang and Zani Xu
This post has been edited 2 times. Last edited by Taco12, Oct 6, 2023, 1:06 AM
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EthanWYX2009
858 posts
#2 • 1 Y
Y by ys-lg
$P(0,0)\Rightarrow f(f(0))=0.$
$P(0,y)\Rightarrow y=f(f(y))+f(f(0))=f(f(y)).$
$\Rightarrow f(2x+f(y))=y-2x.$
$f(2x+f(2y+f(z)))=f(2x+z-2y)=2y+f(z)-2x.$
$t=2x-2y\Rightarrow f(t+z)=f(z)-t.$
$z=0\Rightarrow f(t)=f(0)-t=c_1-t.$$\Rightarrow \forall 2\mid x,f(x)=c_1-x.$
$z=1\Rightarrow f(t+1)=f(1)-t.\Rightarrow \forall 2\nmid x,f(x)=c_2-x$
$\Rightarrow f(x)=c-x.\blacksquare$
This post has been edited 2 times. Last edited by EthanWYX2009, Oct 6, 2023, 1:31 AM
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cj13609517288
1908 posts
#3
Y by
@above no ! so close tho

First three steps are same as above. Since $f$ is an involution, $f(y)$ can be $0$ or $1$.
Varying $x$ yields that $f(2x)=C_1-2x$ for some $C_1$ and $f(2x+1)=C_2-(2x+1)$ for some $C_2$.

Finally note that if $C_1$ is even then $C_2$ has to be even, and vice versa. If $C_1$ is odd then $C_2=C_1$. So our solution is $C_1,C_2$ even and $C_1=C_2$ odd. This works upon checking.

Remark
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EthanWYX2009
858 posts
#4 • 4 Y
Y by cj13609517288, BigJoJo, LLL2019, ys-lg
I didn't check the final result....... I'm so fool
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MathLuis
1524 posts
#5
Y by
The Speedrun begins!.
Let $P(x,y)$ the assertion of the following F.E., i claim that $f(2n)=c-2n, f(2n+1)=c_1-2n-1$ works for all integers $n$ where either $c=c_1$ are odd or $c,c_1$ are both even.
$P(0,x)$
$$f(f(x))+f(f(0))=x \implies f \; \text{bijective}$$Also $P(0,0)$ gives $f(f(0))=0$ so infact $f$ is an involution (i.e. $f(f(x))=x$).
Hence our functional equation became $f(2x+f(y))=y-2x$, now set $f(c)=0$ then by $P(x,c)$ we get that $f(x)=c-x$ for all even $x$.
Also by $P(x,f(1))$ we get $f(2x+1)=(f(1)+1)-2x-1$ for all integers $x$ so $f(x)=c_1-x$ for all odd $x$, now if $c_1$ is odd then notice that $x=f(f(x))=f(c_1-x)=c-c_1+x$ for all odd $x$. so $c=c_1$ odd, if $c_1$ is even, but $c$ odd then $x=f(f(x))=f(c-x)=c_1-c+x$ for all $x$ even so $c=c_1$ which cant happen so $c$ is also even in this case. Hence our claim is true and we are done :D.
This post has been edited 1 time. Last edited by MathLuis, Oct 6, 2023, 1:49 AM
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tadpoleloop
311 posts
#6
Y by
Taco12 wrote:
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $x$ and $y$, \[ f(2x+f(y))+f(f(2x))=y. \]
Calvin Wang and Zani Xu

What an interesting answer.

Claim:
$$\boxed{f(x) = \left\{ \begin{matrix}a-x & x \text{ even}\\b-x & x \text{ odd}\end{matrix}\right. \quad\text{where }a=b\text{ or a,b both even}}$$
Proof:
$P(0,0) \implies f(f(0)) = 0$
$P(0,x)\implies f(f(x)) = x$
$P(x,f(y))\implies f(2x+y) = f(y) - 2x$
In particular $f(2x) = f(0) - 2x$ and $f(2x+1) = (f(1)+1)-(2x+1)$

So $a=f(0)$ and $b = f(1) + 1$ as per our claim.

Plugging into our involution requirement we see that if either $a$ or $b$ are odd that $a=b$ $\square$
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john0512
4187 posts
#7
Y by
The answer is $f(x)=c-x$ for even $x$ and $f(x)=d-x$ for odd $x$ where either $c,d$ are both even or $c=d.$ These solutions clearly work.

Letting $x=y=0$, we get that $f(f(0))=0$. Letting only $x=0$, we have $$f(f(y))+f(f(0))=y$$so $$y=f(f(y)).$$Thus, $f$ is an involution.

Let $f(0)=c$ (and $f(c)=0$).

With this, the original functional equation implies that $$f(2x+f(y))+2x=y.$$Letting $y=2x$ gives $$f(2x+f(2x))=0.$$Since involutions are both injective and surjective, we can de-nest this into $$2x+f(2x)=c$$$$f(2x)=c-2x.$$Thus, for all even $n$, we have $$f(n)=c-n.$$
Plugging in $x=1$, we have that $$f(2+f(y))=y-2.$$Applying $f$ to both sides and using the fact that $f$ is an involution, $$2+f(y)=f(y-2)$$$$f(y)-f(y-2)=-2.$$Thus, by induction, if $f(1)=d-1$, then $$f(odd)=d-odd$$for all odd integers $odd$.

However, note that $$f(f(2x))=f(c-2x)=2x.$$If $c$ is even, then this equation is clearly true since $c-2x$ is even. Then, if $c$ is even, then we have that all even inputs lead to even outputs. Hence, if $d$ is odd, then odd also maps to even, contradicting surjectivity. Thus, if $c$ is even, then $d$ must also be even. Otherwise, if $c$ is odd, then we must have $$d-(c-2x)=2x$$$$d=c.$$Thus we are done.
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v_Enhance
6877 posts
#8 • 2 Y
Y by ESAOPS, Marcus_Zhang
Solution from Twitch Solves ISL:

The answers are \[ f(x) = \begin{cases} a - x & x \equiv 0 \pmod 2 \\ b - x & x \equiv 1 \pmod 2 \end{cases} \]where $a$ and $b$ are either both even, or $a = b$. It can be checked that all of these work, so we prove they're the only solutions.
Let $P(x,y)$ be the given assertion.
  • $P(0,0) \implies f(f(0)) = 0$.
  • $P(0,t) \implies f(f(t)) = t$.
  • $P(1,f(z)) \implies f(z+2)=z-2$.
The last equation $f(z+2) = z-2$ implies $f$ takes the above form for some $a$ and $b$, so we'd be done if we could show the parity condition. If $a$ is odd, then plug in $x=0$ to deduce $a=b$; if $b$ is odd, plug in $x=1$ to deduce $b=a$. This finishes the problem.
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shendrew7
795 posts
#9 • 1 Y
Y by rstenetbg
It's easy to get $f(f(0))=0$ from $(0,0)$ and $f(f(n))=n$ from $(0,n)$. Then our condition can be rewritten as
\[y=f(2x+f(y))+2x.\]
We then substitute values to determine the value of $f$ based on parity:
\begin{align*}
(-t,f(2t)):& \quad f(2t)=-2t+f(0) \implies f(x)=-x+c \text{ for even } x \\
(-t, f(2t+1)):& \quad f(2t+1)=-2t+f(1) \implies f(x)=-x+d \text{ for odd } x
\end{align*}
We finish by using casework on the parities of $c$ and $d$ with our involution $f(f(n))$. We get the following solution, which can be easily tested:
\[\boxed{f(x) = \begin{cases} -x+c & \text{for even } x \\ -x+d & \text{for odd } x \end{cases} \quad \text{where }a,b \text{ both even or } a=b.}\]
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vsamc
3789 posts
#11
Y by
Solution
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eibc
600 posts
#12
Y by
All solutions are of the form $f(x) = -x + c$ for odd $x$ and $f(x) = -x + d$ for even $x$ where $c, d$ are integers such that either $c = d$ or $c \equiv d \equiv 0 \pmod 2$. With some effort, we can verify that these all work.

Denote the original assertion as $P(x, y)$. From $P(0, 0)$ we have $f(f(0)) = 0$. Then from $P(0, y)$, we have $f(f(y)) = y$, which implies that $f$ is bijective. This also lets us rewrite the original equation as $f(2x + f(y)) + 2x = y$.

Then, from $P(1, f(x))$, we have $f(x + 2) + 2 = f(x)$, which implies that there exists integers $c, d$ such that $f(x) = -x + c$ when $x$ is odd and $f(x) = -x + d$ when $x$ is even. From $f(f(0)) = 0$, we see that:
  • If $d$ is even, then $c$ must also be even, as if $c$ is odd then $1 = f(f(1)) = f(-1 + c) = 1 - c + d \equiv 0 \pmod 2$, a contradiction. Any pair $(c, d)$ with $c$ and $d$ both even will work, as mentioned above.
  • If $d$ is odd, then $0 = f(f(0)) = f(d) = -d + c$, so $c = d$, which also works.
Having exhausted all cases, we are done.
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Martin2001
149 posts
#13
Y by
Plug in $P(0,0)$ then $P(0,y)$ to get $f(f(y))=y.$ Thus, $f$ is an involution. The original equation is now
$$y=f(f(y)+2x)+2x.$$Then we can separate into cases even and odd like this :
\begin{align*}
P(-t, f(2t))=f(2t)=-2t+f(0) &\rightarrow f(x)=-x+c \\
p(-t, f(2t+1))=f(2t+1)=-2t+f(1) &\rightarrow f(x)=-x+d.
\end{align*}If $d$ is even, $c$ must also be even, as if $c$ is odd then $1=f(f(1))=f(-1+c)=1-c+d \equiv 0 \pmod 2,$ contradiction.
\newline If $d$ is odd, then $c=d,$ because $0=f(f(0))=f(d)=-d+c,$ which works. Thus, our answer is
$$\boxed{f(x) = \begin{cases} -x+c & \text{for even } x \\ -x+d & \text{for odd } x \end{cases} \quad \text{where }c,d \text{ both even or } c=d.}$$
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Jndd
1416 posts
#14
Y by
We claim that the solution is $f(x)=-x+c$ for even $x$ and $f(x)=-x+d$ for odd $x$ where $c=d$ or $c$ and $d$ are both even. It is easy enough to do cases and check that these solutions work.

Let $P(x,y)$ denote the assertion. $P(0,0)$ gives us $f(f(0))=0$, so by $P(0,y)$, we get \[f(f(y))+f(f(0))=f(f(y))=y,\]meaning $f$ is an involution, and is therefore bijective. Now, We have \[f(2x+f(y))+f(f(2x))=f(2x+f(y))+2x=y,\]giving $f(2x+f(0))=-2x$. Thus, for even $x$, we have $f(x)=-x+f(0)$. We also have $f(2x+f(1))=1-2x$, giving $f(x)=-x+(f(1)+1)$ for odd $x$. Let $c=f(0)$ and $d=f(1)+1$, so we can write $f(x)=-x+c$ for even $x$ and $f(x)=-x+d$ for odd $x$.

Now, suppose $y$ is even in our original equation. Then, we have \[f(2x+f(y))+f(f(2x))=f(2x-y+c)+2x=y,\]so if $c$ is even, then this works. Otherwise, if $c$ is odd, then we must have $c=d$.

Now, suppose $y$ is odd in our original case. Then, we have \[f(2x+f(y))+f(f(2x))=f(2x-y+d)+2x=y,\]so if $d$ is even then this works. Otherwise, we must have $c=d$.

This means that either $c=d$, or $c$ and $d$ are both even, as desired.
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Eka01
204 posts
#15
Y by
I might be missing something cuz I'm sleepy but here goes :-
Put $x=0$ to get $f(f(y))+f(f(0))=y$. Putting $y=0$ gives $f(f(y))=y$. This implies $f$ is bijective.
Now let $a$ be such that $f(a)=0$. Putting $y=a$ gives us that $f(2x)+2x=a$ since $f(f(2x)=2x$ giving us that $f(2x)=a-2x$.
Now let $b$ be such that $f(b)=1$. Putting $y=b$ gives us that $f(2x+1)=b-2x$.

Now using these expressions, we see that either $a$ is even and $b$ is odd or $a+1=b$ in order to satisfy the given equation.
Hence our solution is $\boxed{f(2x)=a-2x, f(2x+1)=b-2x}$ where $a,b$ satisfy the above conditions.
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eg4334
637 posts
#16
Y by
if $P$ is the assertion then $P(0, 0) \implies f(f(0))=0$. Then $P(0, y) \implies f(f(y))=y$. We also have $2x+f(y)=f(y-2x)$ which when $x=1$ gives us $f(y)=f(y-2)-2$. Thus our solutions are determined by $f(0)$ and $f(1)$, say write it as $f(x) = a-x$ when $x$ is odd and $f(x)=b-x$ when $x$ is even. Now $f(f(0)) = 0$ tells us that $f(b) = 0$. Therefore, $b$ can be even or $a=b$. A similar analysis on $f(f(1))=1$ tells us that $a$ is even or $a=b$. Thus, we have $f(x)=a-x$ when $x$ is odd and $f(x)=b-x$ when $x$ is even for some $a, b$ such that they are both even or equal.
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Marcus_Zhang
980 posts
#17
Y by
Storage
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Maximilian113
575 posts
#18
Y by
Let $P(x, y)$ denote the assertion. Then $P(0,0) \implies f(f(0))=0.$ Hence $$P(0, x) \implies f(f(x))=x,$$so $f(x)$ is bijective. Thus the assertion becomes $$P(x, y) \iff f(2x+f(y))=y-2x.$$Thus $$P(x, f(0)) \implies f(2x)=f(0)-2x, P(x, f(1)) \implies f(2x+1)=f(1)-2x.$$If $f(0)$ is odd, we have that $$f(f(2x)) = f(f(0)-2x)=f(1)-f(0)+2x \implies f(1)=f(0).$$This holds also when $f(1)$ is odd. Hence, the solutions are $$f(2x)=a-x, f(2x+1)=b-x$$where either $a, b$ are both even or $a=b.$
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Ilikeminecraft
616 posts
#19
Y by
I claim the answer is $f(x) = -x + c$ if $c$ is some odd integer, or $f(x) = -x + c_1$ if $x$ is even and $f(x) = -x + c_2$ if $x$ is odd, and $c_1, c_2$ are even integers.

Let $P(x, y)$ denote our assertion. From $P(0, 0),$ we see that $f(f(0)) = 0.$ From $P(0, x),$ we get that $f(f(x)) = x,$ using the fact that $f(f(0)) = 0.$ By rearranging our given equation, and taking $f$ on both sides, we see that $2x + f(y) = f(y - 2x).$ Let $f(0) = c_1, f(1) = c_2.$ Note that by plugging in $y = 0, 1$, $f(2x) = c_1 - 2x, f(2x + 1) = -2x + c_2.$ Now we consider two seperate cases.
\begin{enumerate}
\item If $c_1\equiv1\pmod2,$ we plug it back into our original equation to see that $-2x = f(2x + f(0)) = f(2x + c_1),$ and thus we have that $-2x - c_1 + 1 + c_2 = -2x.$ Thus, $1 + c_2 = c_1,$ and hence $f(0) = f(1) + 1.$ Thus, we can write $f(x) = -x + c.$
\item If $c_1 \equiv 0\pmod 2,$ clearly we can't say anything, and both solutions are valid.
\end{enumerate}
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