Found problems: 1148
Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares.
[asy]import olympiad;
import math;
import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin;
B = A + 5 * right;
C = (9/5, 12/5);
pair X = .7 * A + .3 * B;
pair Xa = X + dir(135);
pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C);
pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2;
pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y);
pair Ya2 = (Ya.x, X.y);
pair Yb1 = (Yb.x, X.y);
pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle);
draw(Ya--Ya1--X--Ya2--cycle);
draw(Yb--Yb1--X--Yb2--cycle);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$\mathcal P$", Oa, origin);
label("$\mathcal Q$", Ob, origin);[/asy]
During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.
Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values?
[i]I. Rubanov[/i]
Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.
Johnny writes down quadratic equation
\[ax^2 + bx + c = 0.\]
with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
Let $b \geq 2$ be a positive integer.
(a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits.
(b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits.
(c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even.
(d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8)+fontsize(8));
draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-2.687,0),E);
label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
Let $p,q$ be two distinct odd primes. Calculate
$\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.
Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a
$\frac a2x^2+bx+c=0$ between $r$ and $s$.
$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.
A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
$ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$
Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$
In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]
For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians.
$ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation
$$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.
Find all pairs of real numbers $(x,y)$ satisfying both equations
\[
3x^2+3xy+2y^2 =2
\]
\[
x^2+2xy+2y^2 =1.
\]
[i]2020 CCA Math Bonanza Team Round #5[/i]
If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals
$\textbf{(A) } 0\qquad \textbf{(B) }1 \qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } \text{not uniquely determined}$
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$
Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$