This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 236

1991 AMC 12/AHSME, 20

The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

2003 AMC 12-AHSME, 21

The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero? $ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$

2007 ITest, 18

Tags: vieta , quadratic
Suppose that $x^3+px^2+qx+r$ is a cubic with a double root at $a$ and another root at $b$, where $a$ and $b$ are real numbers. If $p=-6$ and $q=9$, what is $r$? $\textbf{(A) }0\hspace{20.2em}\textbf{(B) }4$ $\textbf{(C) }108\hspace{19.3em}\textbf{(D) }\text{It could be 0 or 4.}$ $\textbf{(E) }\text{It could be 0 or 108.}\hspace{12em}\textbf{(F) }18$ $\textbf{(G) }-4\hspace{19em}\textbf{(H) } -108$ $\textbf{(I) }\text{It could be 0 or }-4.\hspace{12em}\textbf{(J) }\text{It could be 0 or }-108.$ $\textbf{(K) }\text{It could be 4 or }-4.\hspace{11.5em}\textbf{(L) }\text{There is no such value of }r.$ $\textbf{(M) }1\hspace{20em}\textbf{(N) }-2$ $\textbf{(O) }\text{It could be }-2\text{ or }-4.\hspace{10.3em}\textbf{(P) }\text{It could be 0 or }-2.$ $\textbf{(Q) }\text{It could be 2007 or a yippy dog.}\hspace{6.6em}\textbf{(R) }2007$

2004 Romania Team Selection Test, 6

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

2006 IMC, 5

Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.

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.

2013 Korea National Olympiad, 7

Tags: algebra , vieta
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows: \[ b_1 = c_1 = 1 \] \[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \] \[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \] Let $a_k = b_{2014} $. Find the value of \[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]

2013 AMC 10, 19

The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? $\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $

2007 Harvard-MIT Mathematics Tournament, 20

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

2006 Putnam, A5

Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.

2004 AMC 12/AHSME, 23

The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible? $ \textbf{(A)}\ 250,\!000 \qquad \textbf{(B)}\ 250,\!250 \qquad \textbf{(C)}\ 250,\!500 \qquad \textbf{(D)}\ 250,\!750 \qquad \textbf{(E)}\ 251,\!000$

2021 Science ON all problems, 3

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

2004 AMC 12/AHSME, 17

Tags: logarithm , vieta
For some real numbers $ a$ and $ b$, the equation \[ 8x^3 \plus{} 4ax^2 \plus{} 2bx \plus{} a \equal{} 0 \]has three distinct positive roots. If the sum of the base-$ 2$ logarithms of the roots is $ 5$, what is the value of $ a$? $ \textbf{(A)}\minus{}\!256 \qquad \textbf{(B)}\minus{}\!64 \qquad \textbf{(C)}\minus{}\!8 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 256$

2008 Harvard-MIT Mathematics Tournament, 7

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$.

2013 Iran Team Selection Test, 5

Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2010 Bulgaria National Olympiad, 2

Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.

2013 Canada National Olympiad, 1

Determine all polynomials $P(x)$ with real coefficients such that \[(x+1)P(x-1)-(x-1)P(x)\] is a constant polynomial.

1978 AMC 12/AHSME, 13

If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals $\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

2012 Iran MO (2nd Round), 2

Consider the second degree polynomial $x^2+ax+b$ with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, $a^2-4b$ be greater than or equal to zero. Note that the discriminant is also a polynomial with variables $a$ and $b$. Prove that the same story is not true for polynomials of degree $4$: Prove that there does not exist a $4$ variable polynomial $P(a,b,c,d)$ such that: The fourth degree polynomial $x^4+ax^3+bx^2+cx+d$ can be written as the product of four $1$st degree polynomials if and only if $P(a,b,c,d)\ge 0$. (All the coefficients are real numbers.) [i]Proposed by Sahand Seifnashri[/i]

1981 IMO Shortlist, 12

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2009 USA Team Selection Test, 5

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

2018-2019 Winter SDPC, 1

Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

1988 IMO Longlists, 27

Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

2015 Purple Comet Problems, 10

Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$

PEN A Problems, 112

Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.