...math>n\ge1</math>. Let <math>A = (T(2009))^{T(2009)}</math> and <math>B = (T(2009))^A</math>. What is the largest integer <math>k</math> for which <cmat <math>\textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E) ...

Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>. ...

...ximum value of <math>\frac{(2^t-3t)t}{4^t}</math> for real values of <math>t?</math> <math>\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{ ...

...actually, } x^5 - x^4 - x^2 + x = x(x^4 - x^3 - x + 1) \text{? That doesn't factor nicely)}. t ...

...denote the number of red and blue socks, respectively. Also, let <math>t=r+b</math>. The probability <math>P</math> that when two socks are drawn random ...1)}{(r+b)(r+b-1)}+\frac{b(b-1)}{(r+b)(r+b-1)}=\frac{r(r-1)+(t-r)(t-r-1)}{t(t-1)}=\frac{1}{2}. ...

...<math>p</math> does not divide <math>x - t^2</math> for any integer <math>t</math>. {{USAMO newbox|year=2020|num-b=2|num-a=4}} ...

...he complex plane, any spiral similarity can be expressed in the form <math>T(x) = x_0+k (x-x_0),</math> where <math>k</math> is a complex number. The ma ...math> with corresponding complex numbers <math>a', a, b',</math> and <math>b,</math> so ...

real t=2^0.5; D((0,0)--(4*t,0)--(2*t,8)--cycle); ...

3. The fraction, <math>\frac{ab+bc+ac}{(a+b+c)^2}</math>, where <math>a,b</math> and <math>c</math> are side lengths of a triangle, lies in the inter ...th> be a 2-digit [[positive integer]] satisfying <math>\overline{ab}^2=a! +b!</math>. Find the sum of all possible values of <math>\overline{ab}</math> ...

Given that <math>(1+\sin t)(1+\cos t)=5/4</math> and :<math>(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},</math> ...

...*}bc\sin\frac A2\sin\frac B2\sin\frac C2&=bc\cdot\dfrac{(a-b+c)(b-c+a)(c-a+b)}{8abc}\\&=\dfrac{(30-32+34)(32-34+30)(34-30+32)}{8\cdot 32}=\boxed{126}.\e ...the area for <math>[A I_1 I_2]</math> because <math>b, c, \angle A, \angle B, \angle C</math> are all constants while only <math>\sin \alpha</math> is v ...

...s series of geometry competitions named after <b>Igor Fedorovich Sharygin</b> (13/02/1937 – 12/03/2004), a renowned Soviet and Russian mathematician, te ...correspondence round</b> held from January to March, and a <b>final round</b>. The competition is open to Russian students in grades 8 to 11 and is also ...

...<math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>. ...th>MA = \frac{\sqrt{2}}{2} \approx 0.707 > \frac{3}{5}</math>, point <math>B</math> lies between <math>M</math> and <math>A</math> on the semicircular a ...

of angles ∠bac, ∠acb, and ∠cba are A, B, and C, respectively, and the perimeter of the triangle with sides AB, BC, b + c, ...

...h> of <math>\Omega</math> with <math>|B|=|A|</math>, it holds that <math>f(B) \geq f(A)</math>. \forall B:(B \subseteq \Omega, \# B=\# A) \rightarrow \mu(B) \geq \mu(A) . ...

...h> of <math>\Omega</math> with <math>|B|=|A|</math>, it holds that <math>f(B) \geq f(A)</math>. \forall B:(B \subseteq \Omega, \# B=\# A) \rightarrow \mu(B) \geq \mu(A) . ...

...i+1 = (x_i-1)^2 = 1-P \Longleftrightarrow x_i = 1 \pm t</math> where <math>t = \pm \sqrt{1-P} \in \mathbb{R}</math>. If <math>t=0</math>, then we have <math>(x_1,x_2,x_3,x_4)=(1,1,1,1)</math> which is a ...

...\frac{x^{2}}{t-1}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\frac{w^{2}}{t-7^{2}}=1.</cmath> ...6, 36, 64\}</math>. After clearing fractions, for each of the values <math>t=4,16,36,64</math>, we have the equation ...

...t</math>. Show that <math>k=a^2-3b\ge 0</math> and that <math>\sqrt k\le r-t</math>. By Vieta's Formulas, <math>a=-r-s-t</math>, <math>b=rs+st+rt</math>, and <math>c=-rst</math>. ...

...top two vertices of <math>T</math> are in <math>R \cup S</math>, and <math>T</math> contains <math>\frac{1}{4}</math> of the lattice points contained in label(scale(1.3)*"$T$",(3.5,25/2)); ...