Difference between revisions of "User:Ddk001"

Latest revision as of 15:25, 11 August 2025

If you have a problem or solution to contribute, please go to this page.

I am a aops user who likes making and doing problems, doing math, and redirecting pages (see Principle of Insufficient Reasons). I like geometry and don't like counting and probability. My number theory skill are also not bad

User Count

If this is your first time visiting this page, edit it by incrementing the user count below by one.

$\sqrt{1849}$

Please do not mess with the user count. I would know and I would find you.

Credits given to Firebolt360 for inventing the box above.

Cool asyptote graphs

Asymptote is fun! $[asy]draw((0,0)----(0,6));draw((0,-3)----(-3,3));draw((3,0)----(-3,6));draw((6,-6)----(-6,3));draw((6,0)----(-6,0));[/asy]$

$[asy]draw(circle((0,0),1));draw((1,0)----(0,1));draw((1,0)----(0,2));draw((0,-1)----(0,2));draw(circle((0,3),2));draw(circle((0,4),3));draw(circle((0,5),4));draw(circle((0,2),1));draw((0,9)----(0,18));[/asy]$

Contributions

AMC

2022 AMC 12B Problems/Problem 25 Solution 5 (Now it's solution 6)

2023 AMC 12B Problems/Problem 20 Solution 3

AIME

2016 AIME I Problems/Problem 10 Solution 3

2017 AIME I Problems/Problem 14 Solution 2

2019 AIME I Problems/Problem 15 Solution 6

2022 AIME II Problems/Problem 3 Solution 3

Restored diagram for 1994 AIME Problems/Problem 7

USAMO/USAJMO

1978 USAMO Problems/Problem 1 Solution 4

2002 USAMO Problems/Problem 2 Solution 2

Restored solution for 2022 USAMO Problems/Problem 6 Solution 1 See https://artofproblemsolving.com/wiki/index.php?title=2022_USAMO_Problems/Problem_6&action=history

IMO

1995 IMO Problems/Problem 6 Solution 1 (I got the solution from a forum discussion)

1986 IMO Problems/Problem 3 Solution 2 (I may or may not had gotten the solution from another source)

1976 IMO Problems/Problem 6 Solution 2 (I actually made this solution by myself! :) )

Problems Sharing Contest

Here, you can post all the math problems that you have. Everyone will try to come up with a appropriate solution. The person with the first solution will post the next problem. I'll start:

1. There is one and only one perfect square in the form

$(p^2+1)(q^2+1)-((pq)^2-pq+1)$

where $p$ and $q$ are prime. Find that perfect square. ~Ddk001

$\textbf{Solution by cxsmi}$

1. We can expand the product in the expression. $(p^2+1)(q^2+1)-((pq)^2-pq+1) = p^2q^2+p^2+q^2+1-((pq)^2-pq+1) = p^2 + q^2 + pq$ . Suppose this equals $m^2$ for some positive integer $m$ . We rewrite using the square of a binomial pattern to find that $m^2 = (p + q)^2 - pq$ . Through trial and error on small values of $p$ and $q$ , we find that $p$ and $q$ must equal $3$ and $5$ in some order. The perfect square formed using these numbers is $\boxed{49}$ .

Note: I will be the first to admit that this solution is somewhat lucky.

2. A diamond is created by connecting the points at which a square circumscribed around the incircle of an isosceles right triangle $\triangle ABC$ intersects $\triangle ABC$ itself. $\triangle ABC$ has leg length $2024$ . The perimeter of this diamond is expressible as $a\sqrt{b}-c$ , where $a$ , $b$ , and $c$ are integers, and $c$ is not divisible by the square of any prime. What is the remainder when $a + b + c$ is divided by $1000$ ?

$[asy] unitsize(1inch); draw((0,0)--(0,2)); draw((0,2)--(2,0)); draw((2,0)--(0,0)); draw(circle((0.586,0.586),0.586)); draw((0,0)--(0,1.172),red); draw((0,1.172)--(1.172,1.172)); draw((1.172,1.172)--(1.172,0)); draw((1.172,0)--(0,0),red); draw((0,1.172)--(0.828,1.172),red); draw((0.828,1.172)--(1.172,0.828),red); draw((1.172,0.828)--(1.172,0),red); draw((0,0.1)--(0.1,0.1)); draw((0.1,0.1)--(0.1,0)); label("$A$",(0,2.1)); label("$B$",(0,-0.1)); label("$C$",(2,-0.1)); label("$2024$",(-0.2,1)); label("$2024$",(1,-0.2)); [/asy]$

$\textbf{Solution by Ddk001}$

Solution 1

The inradius of $\Delta ABC$ , $r$ , can be calculated as

$r=\frac{\textbf{Area}_{ABC}}{\textbf{Semiperimeter}} \implies r=\frac{2024^2/2}{(2024+2024+2024 \sqrt{2})/2}=\frac{2024}{2+\sqrt{2}}=2024-1012\sqrt{2}$

so the square have side length $4048-2024 \sqrt{2}$ . Let the $D$ be the vertex of the square $D \ne B$ on side $BC$ . Then $DC= 2024 (\sqrt{2} -1)$ . Let the sides of the square intersect $AC$ at $E$ and $F$ , with $AE<AF$ . Then $AE=CF=2024(2-\sqrt{2})$ so $EF=2024 (3 \sqrt{2} -4)$ . Let $G$ be the vertex of the square across from $B$ . Then $EG=FG=2024 (3-2\sqrt{2})$ . Thus the perimeter of the diamond is

$4(4048-2024 \sqrt{2})-2 \cdot 2024 (3-2\sqrt{2})+2024 (3 \sqrt{2} -4)=2024 (8-4 \sqrt{2}-6+4\sqrt{2}+3\sqrt{2}-4)=2024(3\sqrt{2}-2)=7072\sqrt{2}-4048$

The desired sum is $7072+2+4048=\boxed{11122}$ . $\blacksquare$

Ddk001 Presents

THE FOLLOWING PROBLEM

Note: This is one of my favorite problems. Very well designed and actually used two of my best tricks without looking weird.

Suppose

$x \equiv 2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6 \pmod{7!}$

Find the remainder when $\min{x}$ is divided by 1000.

Vandalism area

Congratulations! You had reached the end of this boring user page. Time for vandalism! Write anything under 45000 bytes.

Hi Ddk001 ~ zhenghua

Today my math teacher at school thought that a triangle had to be equilateral just because an altitude could be drawn in it and graded my homework 90% because the wrong answer was correct.... (Wait can't any triangle have an altitude?)

Vandalism Page

@@ Line 9: / Line 9: @@
 ==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==
 <div style="margin-left: 10px; margin-bottom:10px"><font color="black">If this is your first time visiting this page, edit it by incrementing the user count below by one.</font></div>
-<center><font size="100px">37</font></center>
+<center><font size="100px"><math>\sqrt{1849}</math></font></center>
+<div style="margin-left: 10px; margin-bottom:10px"><font color="black">Please do not mess with the user count. I would know and I would find you.</font></div>
 </div>
@@ Line 25: / Line 26: @@
 ==Contributions==
-[[2005 AMC 8 Problems/Problem 21]] Solution 2
+===AMC===
 [[2022 AMC 12B Problems/Problem 25]] Solution 5 (Now it's solution 6)
 [[2023 AMC 12B Problems/Problem 20]] Solution 3
+===AIME===
 [[2016 AIME I Problems/Problem 10]] Solution 3
@@ Line 38: / Line 42: @@
 [[2022 AIME II Problems/Problem 3]] Solution 3
+Restored diagram for [[1994 AIME Problems/Problem 7]]
+===USAMO/USAJMO===
 [[1978 USAMO Problems/Problem 1]] Solution 4
-Restored diagram for [[1994 AIME Problems/Problem 7]]
+[[2002 USAMO Problems/Problem 2]] Solution 2
+Restored solution for [[2022 USAMO Problems/Problem 6]] Solution 1
+See https://artofproblemsolving.com/wiki/index.php?title=2022_USAMO_Problems/Problem_6&action=history
+===IMO===
+[[1995 IMO Problems/Problem 6]] Solution 1 (I got the solution from a forum discussion)
+[[1986 IMO Problems/Problem 3]] Solution 2 (I may or may not had gotten the solution from another source)
+[[1976 IMO Problems/Problem 6]] Solution 2 (I actually made this solution by myself! :) )
+===Theorems===
 [[Divergence Theorem]]
@@ Line 49: / Line 70: @@
 [[Principle of Insufficient Reasons]]
-[[Nested Sum Theorem]]
+[[The Nested Sum Theorem]]
+[[Hensel's Lemma]]
+===Definitions===
+[[Laplace Transform]]
+[[Polynomial Congruences]]
+[[Primitive Roots]]
+[[Order of a integer]]
+[[Index]]
+[[Diagonalizability]]
+===Others===
 [[Proofs to Some Number Theory Facts]]
-[[Hensel's Lemma]]
+[[Chebyshev and the history of the Prime Number Theorem]]
+[[Proof of the Existence of Primitive Roots]]
+[[Proof of Some Primitive Roots Facts]]
 ==Problems Sharing Contest==
@@ Line 125: / Line 168: @@
 ==Vandalism area==
-Here, you can add anything, delete anything, and do anything! (Don't delete this line since it's instruction and don't be inappropriate) Do not delete the see also. However, do NOT vandalize before this word (Feel free to delete this and the period that follows).
+Congratulations! You had reached the end of this boring user page. Time for vandalism! Write anything under 45000 bytes.
-Okay
-(ok :) :) this page is so cool!)
-honestly i think your user page is very cool. :)
+Hi Ddk001 ~ zhenghua
-Hi Ddk001  [[User:zhenghua]] (Taking Oly Geo)
+Today my math teacher at school thought that a triangle had to be equilateral just because an altitude could be drawn in it and graded my homework 90% because the wrong answer was correct....
+(Wait can't any triangle have an altitude?)
-Zhenghua I havent seen you since forever!!! I'm not focusing on compeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeetition right now so you probably won't see me in any of your classes.
+[[Vandalism Page]]
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