Found problems: 236
2016 Indonesia TST, 2
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
PEN A Problems, 1
Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.
2010 Romania National Olympiad, 1
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
2002 AMC 12/AHSME, 13
Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$?
\[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3
\]
2014 NIMO Problems, 5
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
[i]Proposed by Justin Stevens[/i]
2013 NIMO Problems, 5
Let $x,y,z$ be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z \\
y^2 + 5z &= 10y \\
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of $z$.
[i]Proposed by Aaron Lin[/i]
2008 Hong kong National Olympiad, 2
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
2020 LIMIT Category 1, 7
Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$
(A)$5$
(B)$6$
(C)$7$
(D)$8$
2020/2021 Tournament of Towns, P2
Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?
2002 AMC 12/AHSME, 6
Suppose that $ a$ and $ b$ are are nonzero real numbers, and that the equation $ x^2\plus{}ax\plus{}b\equal{}0$ has solutions $ a$ and $ b$. Then the pair $ (a,b)$ is
$ \textbf{(A)}\ (\minus{}2,1) \qquad
\textbf{(B)}\ (\minus{}1,2) \qquad
\textbf{(C)}\ (1,\minus{}2) \qquad
\textbf{(D)}\ (2,\minus{}1) \qquad
\textbf{(E)}\ (4,4)$
2010 Math Prize For Girls Problems, 20
What is the value of the sum
\[
\sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]
where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?
2008 Harvard-MIT Mathematics Tournament, 6
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
2014 AIME Problems, 5
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
1967 AMC 12/AHSME, 17
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that:
$\textbf{(A)}\ |r_1+r_2|>4\sqrt{2}\qquad
\textbf{(B)}\ |r_1|>3 \; \text{or} \; |r_2| >3 \\
\textbf{(C)}\ |r_1|>2 \; \text{and} \; |r_2|>2\qquad
\textbf{(D)}\ r_1<0 \; \text{and} \; r_2<0\qquad
\textbf{(E)}\ |r_1+r_2|<4\sqrt{2}$
1970 AMC 12/AHSME, 11
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is
$\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$
1977 AMC 12/AHSME, 27
There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is
$\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$
$\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$
1998 All-Russian Olympiad, 1
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
2012 ELMO Shortlist, 6
Prove that if $a$ and $b$ are positive integers and $ab>1$, then
\[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.
[i]Calvin Deng.[/i]
1989 Federal Competition For Advanced Students, P2, 5
Find all real solutions of the system:
$ x^2\plus{}2yz\equal{}x,$
$ y^2\plus{}2zx\equal{}y,$
$ z^2\plus{}2xy\equal{}z.$
2015 AMC 10, 16
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
2014 Singapore Senior Math Olympiad, 1
If $\alpha$ and $\beta$ are the roots of the equation $3x^2+x-1=0$, where $\alpha>\beta$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$.
$ \textbf{(A) }\frac{7}{9}\qquad\textbf{(B) }-\frac{7}{9}\qquad\textbf{(C) }\frac{7}{3}\qquad\textbf{(D) }-\frac{7}{3}\qquad\textbf{(E) }-\frac{1}{9} $
1999 AIME Problems, 3
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.
2000 AMC 12/AHSME, 22
The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) \equal{} x^4 \plus{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$. Which of the following is the smallest?
$ \textbf{(A)}\ P( \minus{} 1)$
$ \textbf{(B)}\ \text{The product of the zeros of }P$
$ \textbf{(C)}\ \text{The product of the non \minus{} real zeros of }P$
$ \textbf{(D)}\ \text{The sum of the coefficients of }P$
$ \textbf{(E)}\ \text{The sum of the real zeros of }P$
[asy]
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real f(real x) {
real y=1/4;
return 0.2125(x*y)^4-0.625(x*y)^3-1.6125(x*y)^2+0.325(x*y)+5.3;
}
draw(graph(f,-10.5,19.4));
draw((-13,0)--(22,0)^^(0,-10.5)--(0,15));
int i;
filldraw((-13,10.5)--(22,10.5)--(22,20)--(-13,20)--cycle,white, white);
for(i=-3; i<6; i=i+1) {
if(i!=0) {
draw((4*i,0)--(4*i,-0.2));
label(string(i), (4*i,-0.2), S);
}}
for(i=-5; i<6; i=i+1){
if(i!=0) {
draw((0,2*i)--(-0.2,2*i));
label(string(2*i), (-0.2,2*i), W);
}}
label("0", origin, SE);[/asy]
2006 Princeton University Math Competition, 3
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?
2023 USAMTS Problems, 4
Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]