Olympiad problems

2022 Thailand Online MO, 1

Tags: algebra , equation

Determine, with proof, all triples of real numbers $(x,y,z)$ satisfying the equations $$x^3+y+z=x+y^3+z=x+y+z^3=-xyz.$$

Russian TST 2017, P2

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2024 ISI Entrance UGB, P2

Tags: algebra , polynomial , inequalities

Suppose $n\ge 2$. Consider the polynomial \[Q_n(x) = 1-x^n - (1-x)^n .\] Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.

1990 IMO Longlists, 88

Tags: cauchy schwarz , holder , inequality , 4-variable inequality , algebra

Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.

2016 CCA Math Bonanza, L2.4

Tags: algebra , polynomial

What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$? [i]2016 CCA Math Bonanza Lightning #2.4[/i]

2012 Turkmenistan National Math Olympiad, 2

Tags: polynomial , algebra

If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.

2009 District Olympiad, 2

Tags: algebra , system of equations

Real numbers $a, b, c, d, e$, have the property $$|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a|.$$ Prove they are all equal.

VMEO II 2005, 1

Tags: algebra , inequalities

Let $a, b, c$ be three positive real numbers. a) Prove that there exists a unique positive real number $d$ that satisfies $$\frac{1}{a + d}+ \frac{1}{b + d}+\frac{1}{c + d}=\frac{2}{d} .$$ b) With $x, y, z$ being positive real numbers such that $ax + by + cz = xyz$, prove the inequality $$x + y + z \ge \frac{2}{d}\sqrt{(a + d)(b + d)(c + d)}.$$

2022 Princeton University Math Competition, 7

Tags: algebra

Pick $x, y, z$ to be real numbers satisfying $(-x+y+z)^2-\frac13 = 4(y-z)^2$, $(x-y+z)^2-\frac14 = 4(z-x)2$, and $(x+y-z)^2 -\frac15 = 4(x-y)^2$. If the value of $xy+yz +zx$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

2014 USA TSTST, 4

Tags: algebra , polynomial , vector , floor function , modular arithmetic , pigeonhole principle , linear algebra

Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that: (i) both $A$ and $B$ have degree at most $d/2$ (ii) at most one of $A$ and $B$ is the zero polynomial. (iii) $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.

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 ?

1996 Chile National Olympiad, 7

Tags: algebra

(a) Let $a, b, c, d$ be integers such that $ad\ne bc$. Show that is always possible to write the fraction $\frac{1}{(ax+b)(cx+d)}$in the form $\frac{r}{ax+b}+\frac{s}{cx+d}$ (b) Find the sum $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}$$

1999 Brazil Team Selection Test, Problem 2

Tags: algebra , polynomial

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

2009 AMC 12/AHSME, 19

Tags: algebra , polynomial , search , vieta , prime number , number theory

For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

1985 Bulgaria National Olympiad, Problem 1

Tags: polynomial , number theory , algebra

Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)

1996 AIME Problems, 5

Tags: algebra , polynomial

Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a, b,$ and $c,$ and that the roots of $x^3+rx^2+sx+t=0$ are $a+b, b+c,$ and $c+a.$ Find $t.$

2014 ELMO Shortlist, 3

Tags: algebra , polynomial , induction , binomial theorem , number theory

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

1991 IMTS, 2

Tags: algebra

Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares.

2020 CMIMC Algebra & Number Theory, 1

Tags: algebra , number theory

Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.

2011 Hanoi Open Mathematics Competitions, 6

Tags: system of equations , algebra

Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 2 \\ x^4 - y^4 = 5x - 3y \end{cases}$

MathLinks Contest 3rd, 1

Tags: algebra

Find all functions$ f, g : (0,\infty) \to (0,\infty)$ such that for all $x > 0$ we have the relations: $f(g(x)) = \frac{x}{xf(x) - 2}$ and $g(f(x)) = \frac{x}{xg(x) - 2}$ .

1983 AMC 12/AHSME, 18

Tags: algebra , polynomial , function

Let $f$ be a polynomial function such that, for all real $x$, \[f(x^2 + 1) = x^4 + 5x^2 + 3.\] For all real $x$, $f(x^2-1)$ is $ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $

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 $a,b$.

2019 India PRMO, 10

Tags: algebra , clock , time

One day I went for a walk in the morning at $x$ minutes past $5'O$ clock, where $x$ is a 2 digit number. When I returned, it was $y$ minutes past $6'O$ clock, and I noticed that (i) I walked for exactly $x$ minutes and (ii) $y$ was a 2 digit number obtained by reversing the digits of $x$. How many minutes did I walk?

2010 CHMMC Fall, 7

Tags: algebra

Art and Kimberly build flagpoles on a level ground with respective heights $10$ m and $15$ m, separated by a distance of $5$ m. Kimberly wants to move her flagpole closer to Art’s, but she can only doing so in the following manner: 1. Run a straight wire from the top of her flagpole to the bottom of Art’s. 2. Run a straight wire from the top of Art’s flagpole to the bottom of hers. 3. Build the flagpole to the point where the wires meet. If Kimberly keeps moving her flagpole in this way, compute the number of flagpoles she will build whose heights are $1$ m or greater (not counting her original $15$ m flagpole).