| 2 || drabdude || 129 || 57464.853 || 445.464 | 2 || kckk || 30 || 18162.240 || 605.408 ...

[[File:Tur 2 2025 8 1.png|280px|right]] ==2025 II Problem 8 2== ...

In the second iteration, i = 2, so Python prints 2. '''Find <math>\boldsymbol{\sum_{n=1}^{50} 2^{n}.}</math>''' ...

pair M = (2, 0); ...} = \frac{2ZA}{4 - ZA}</math>, yielding <math>ZA = 1</math> and <math>XM = 2</math>. Thus, ...

...margins using the geometry package, use the line <code>\usepackage[margin=2.5cm]{geometry}</code>. Check out the [http://tug.ctan.org/tex-archive/macro \[ \Big(2\sqrt{\frac{a}{b}}\Big)^m + \Big(2\sqrt{\frac{b}{a}}\Big)^m ...

...h>EJCI</math> to the area of one of the faces of the cube. What is <math>R^2?</math> D = (2,0.5); ...

This is my first image. Hello world! This is my first image. ...

...cient Reasons]]). I like geometry and don't like counting and probability. My number theory skill are also not bad ...ircle((0,3),2));draw(circle((0,4),3));draw(circle((0,5),4));draw(circle((0,2),1));draw((0,9)----(0,18));</asy> ...

Let <math>n</math> be the least positive integer for which <math>149^n-2^n</math> is divisible by <math>3^3\cdot5^5\cdot7^7.</math> Find the number ...>. It also shows that <cmath>7=v_7(149^n-2^n) = v_7(n) + v_7(147) = v_7(n)+2</cmath> so thus, <math>7^5</math> divides <math>n</math>. ...

...rt5</math> and <math>\angle AEC = 45^{\circ}.</math> What is <math>CE^2+DE^2?</math> A = (-5*sqrt(2),0); ...

...a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two ===Problem 2=== ...

...\sqrt{3} \qquad\textbf{(B)}\; 18+6\sqrt{3} \qquad\textbf{(C)}\; 12+12\sqrt{2} \qquad\textbf{(D)}\; 30 \qquad\textbf{(E)}\; 32</math> A=(2,0); ...

Draw((2,0)--(2,3), linewidth(1)); Draw((0,2)--(3,2), linewidth(1)); ...

...faultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R ...verline{BD},</math> the longer diagonal of <math>ABCD.</math> Then <math>d^2</math> can be written in the form <math>m+n\sqrt p,</math> where <math>m,n, ...

==Problem 2== ...which comes out to be around <math>15</math> feet. He then measures <math>2</math> feet to either side of the first and last trees in the row before th ...

...nswered before the clock came up around 9 times, the moderator only picked my answer 3 times. Answering before the clock is not enough to guarantee a poi 2 aaravdodhia ...

<cmath> f(x^2)-f(y^2)=(x+y)(f(x)-f(y)) </cmath> ...math>, and by letting <math>x = 0, y = -n</math>, we have <math>f(0) - f(n^2) = -n (f(0) - f(-n))</math>. Substitution and simplification results in ...

...tute <math>y = \sin^2{x}</math>. Since we know that <math>\cos^2{x}=1-\sin^2{x}</math>, we can do some simplification. ...ain to get some cancellation through binomials. If we let <math>z=\frac{1}{2}-y</math>, we can simplify the equation to: ...

...>BC = 22</math>, and <math>TX^2 + TY^2 + XY^2 = 1143</math>. Find <math>XY^2</math>. ...h>, and the cosine law in <math>\triangle TXY</math> gives <cmath>1143-2XY^2=-\frac{11}{8}\cdot XT\cdot YT.</cmath> ...

...cle and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be points on th unitsize(2.5cm); ...