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Quadratic Residues
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Beautiful problem
luutrongphuc   3
N 19 minutes ago by aidenkim119
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
3 replies
luutrongphuc
Apr 4, 2025
aidenkim119
19 minutes ago
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Geometry
youochange   0
22 minutes ago
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
0 replies
youochange
22 minutes ago
0 replies
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comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.
parmenides51   1
N 31 minutes ago by Mathzeus1024
Source: 2013 Cuba 2.9
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
1 reply
parmenides51
Sep 20, 2024
Mathzeus1024
31 minutes ago
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Fridolin just can't get enough from jumping on the number line
Tintarn   2
N 38 minutes ago by Sadigly
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 1
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?
2 replies
Tintarn
Mar 17, 2025
Sadigly
38 minutes ago
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Geometry
Captainscrubz   2
N an hour ago by MrdiuryPeter
Source: Own
Let $D$ be any point on side $BC$ of $\triangle ABC$ .Let $E$ and $F$ be points on $AB$ and $AC$ such that $EB=ED$ and $FD=FC$ respectively. Prove that the locus of circumcenter of $(DEF)$ is a line.
Prove without using moving points :D
2 replies
Captainscrubz
3 hours ago
MrdiuryPeter
an hour ago
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inequality ( 4 var
SunnyEvan   4
N an hour ago by SunnyEvan
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
4 replies
SunnyEvan
Apr 4, 2025
SunnyEvan
an hour ago
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Find the constant
JK1603JK   1
N an hour ago by Quantum-Phantom
Source: unknown
Find all $k$ such that $$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$forall $a,b,c\ge 0.$
1 reply
JK1603JK
4 hours ago
Quantum-Phantom
an hour ago
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2025 - Turkmenistan National Math Olympiad
A_E_R   4
N an hour ago by NODIRKHON_UZ
Source: Turkmenistan Math Olympiad - 2025
Let k,m,n>=2 positive integers and GCD(m,n)=1, Prove that the equation has infinitely many solutions in distict positive integers: x_1^m+x_2^m+⋯x_k^m=x_(k+1)^n
4 replies
A_E_R
2 hours ago
NODIRKHON_UZ
an hour ago
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hard problem
Cobedangiu   15
N an hour ago by Nguyenhuyen_AG
problem
15 replies
Cobedangiu
Mar 27, 2025
Nguyenhuyen_AG
an hour ago
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9x9 board
oneplusone   8
N an hour ago by lightsynth123
Source: Singapore MO 2011 open round 2 Q2
If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.
8 replies
oneplusone
Jul 2, 2011
lightsynth123
an hour ago
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Function L
goldeneagle   7
N Jan 15, 2018 by ThE-dArK-lOrD
Source: Iran 3rd round 2013 - Number Theory Exam - Problem 5
$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow:
$L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$
a) For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points)

b) Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant.
Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.$ . (7 points)

c) Prove that $a+b+c = -3$. (4 points)

d) Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points)

e) Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points)

(${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)
7 replies
goldeneagle
Sep 11, 2013
ThE-dArK-lOrD
Jan 15, 2018
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Function L
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Quadratic Residues
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Source: Iran 3rd round 2013 - Number Theory Exam - Problem 5
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goldeneagle
240 posts
goldeneagle
#1 h  Sep 11, 2013, 2:59 PM • 2 Y
Y by Adventure10, Mango247
$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow:
$L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$
a) For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points)

b) Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant.
Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix} a & & &x \in A \\ b& & &x \in B \\ c& & & x \in C \end{matrix}\right.$ . (7 points)

c) Prove that $a+b+c = -3$. (4 points)

d) Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points)

e) Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points)

(${\mathbb Z_p}^{*} = \mathbb Z_p \setminus \{0\}$)
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qua96
119 posts
qua96
#2 h  Sep 17, 2013, 2:51 AM • 2 Y
Y by Adventure10, Mango247
Jacobstal theorem :)
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odnerpmocon
455 posts
odnerpmocon
#3 h  Sep 17, 2013, 1:50 PM • 1 Y
Y by Adventure10
qua96 wrote:
Jacobstal theorem :)

More details please?
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sahadian
112 posts
sahadian
#4 h  Sep 18, 2013, 3:10 AM • 1 Y
Y by Adventure10
a) $\left ( \frac{t^4}{p} \right )=1$ so we have:
$ L(m) =\sum_{x\in\mathbb{Z}_p}^{ }\left (\frac{x(x^3+m)}{p}\right ) =\sum_{x\in\mathbb{Z}_p}^{ }\left (\frac{x(x^3+m)}{p}\right )\left ( \frac{t^4}{p} \right ) = \sum_{x\in\mathbb{Z}_p}^{ }\left (\frac{xt((xt)^3+mt^3)}{p}\right )$
then $p=3k+1$ so $L(mt^3)= \sum_{x\in\mathbb{Z}_p}^{ }\left (\frac{xt((xt)^3+mt^3)}{p}\right )$
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wiseman
216 posts
wiseman
#5 h  Sep 29, 2014, 3:50 PM • 1 Y
Y by Adventure10
Any ideas for parts b) , c) , d) and e) ??
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Gems98
203 posts
Gems98
#6 h  Feb 12, 2017, 8:54 PM • 1 Y
Y by Adventure10
I think part (b) can use results from part (a) because when $p=3k+1$, the equation $x^3\equiv 1(mod p)$ has 3 answer.
So, set $X=\{d \mid \exists x\in \mathbb{Z} , x^3\equiv d (modp)\}$ has exactly $\frac{p-1}{3}$ elements.
We can choose $e,f \in \mathbb{Z}$ such that $Y=\{ed \mid \exists x\in \mathbb{Z} , x^3\equiv d (modp)\} \cap X = {\o}$,
$Y=\{fd \mid \exists x\in \mathbb{Z} , x^3\equiv d (modp)\} \cap X = {\o}$ and $Y \cap Z= {\o}$
And from part (a), we can partition $\mathbb{Z}/p\mathbb{Z}=X \cup Y \cup Z$
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ThE-dArK-lOrD
4071 posts
ThE-dArK-lOrD
#9 h  Jan 15, 2018, 6:11 PM • 3 Y
Y by Etemadi, Adventure10, Mango247
b)
c)
d)
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ThE-dArK-lOrD
4071 posts
ThE-dArK-lOrD
#10 h  Jan 15, 2018, 6:14 PM • 1 Y
Y by Adventure10
e)
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