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.

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Found problems: 117

2010 Belarus Team Selection Test, 8.1
Tags: sum of digits , number theory , root
The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$. a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots. b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots. (I. Voronovich)

2013 Hanoi Open Mathematics Competitions, 12
Tags: quadratic , inequalities , root , algebra
If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

1971 IMO Longlists, 31
Tags: equation , root , polynomial , algebra
Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

1970 IMO Shortlist, 11
Tags: algebra , polynomial , root
Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$

1988 IMO Shortlist, 16
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2006 IMO, 5
Tags: algebra , polynomial , root
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

2016 Irish Math Olympiad, 3
Tags: algebra , polynomial , root , sum
Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?

1967 IMO Shortlist, 1
Tags: algebra , equation , root , exponential equation
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

1988 IMO, 1
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2020 German National Olympiad, 3
Tags: polynomial , root , algebra , inequalities
Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

2006 MOP Homework, 6
Tags: polynomial , root , algebra
Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial $P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.

2020 Brazil Undergrad MO, Problem 6
Tags: function , quadratic , root , fixed point
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

1952 Moscow Mathematical Olympiad, 221
Tags: algebra , root , trinomial
Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.

2021 Hong Kong TST, 2
Tags: algebra , polynomial , root
Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If \[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\] prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$. (Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)

1988 IMO Longlists, 42
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

1953 Moscow Mathematical Olympiad, 253
Tags: inequalities , root , trinomial , algebra
Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

2012 Dutch BxMO/EGMO TST, 1
Tags: trinomial , quadratic , polynomial , root , algebra
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

1973 IMO, 3
Tags: algebra , polynomial , root , minimum value , coefficient
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

1966 IMO Shortlist, 35
Tags: algebra , polynomial , irrational number , root
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

1976 Euclid, 8
Tags: root , equation algebra
Source: 1976 Euclid Part A Problem 8 ----- Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is $\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E) } \text{none of these}$

1949 Putnam, A5
Tags: root
How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

1941 Moscow Mathematical Olympiad, 080
Tags: trigonometry , analysis , algebra , root
How many roots does equation $\sin x = \frac{x}{100}$ have?

2021 German National Olympiad, 1
Tags: quadratic , quadratic equation , root , algebra
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.

1966 IMO Shortlist, 48
Tags: algebra , equation , number theory , root , parametric equation
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

1985 Spain Mathematical Olympiad, 7
Tags: polynomial , root , integer polynomial , polynomial equation , algebra
Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.