Olympiad problems

2014 Harvard-MIT Mathematics Tournament, 4

Tags: algebra , polynomial , hmmt , vieta , sum of roots

Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.

1979 IMO Longlists, 56

Tags: function , floor function , algebra , diophantine equation

Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.

1992 India National Olympiad, 6

Tags: polynomial , integration , calculus , algebra

Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.

2017 Brazil Team Selection Test, 1

Tags: fraction , algebra

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

2001 China Team Selection Test, 3

Tags: polynomial , algebra , function , functional equation

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

2023 Moldova EGMO TST, 4

Tags: prime number , system of equations , number theory , algebra

Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

1999 Czech And Slovak Olympiad IIIA, 6

Tags: system of equations , algebra

Find all pairs of real numbers $a,b$ for which the system of equations $$ \begin{cases} \dfrac{x+y}{x^2 +y^2} = a \\ \\ \dfrac{x^3 +y^3}{x^2 +y^2} = b \end{cases}$$ has a real solution.

2008 South East Mathematical Olympiad, 2

Tags: algebra

Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$.

2023 Junior Balkan Team Selection Tests - Romania, P1

Tags: algebra

Determine the real numbers $x$, $y$, $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$

2023 Purple Comet Problems, 2

Tags: algebra

There are positive real numbers $a$, $b$, $c$, $d$, and $p$ such that $a$ is $62.5\%$ of $b$, $b$ is $64\%$ of $c$, c is $125\%$ of $d$, and $d$ is $p\%$ of $a$. Find $p$.

2017 Vietnam National Olympiad, 2

Tags: algebra , polynomial

Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} \]

2010 China National Olympiad, 3

Tags: algebra , inequalities , trigonometry , complex number , calculus , function

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

2014 Regional Competition For Advanced Students, 1

Tags: equation , algebra

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

2005 ISI B.Stat Entrance Exam, 3

Tags: function , algebra

Let $f$ be a function defined on $\{(i,j): i,j \in \mathbb{N}\}$ satisfying (i) $f(i,i+1)=\frac{1}{3}$ for all $i$ (ii) $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$ for all $k$ such that $i <k<j$. Find the value of $f(1,100)$.

2020 Moldova Team Selection Test, 6

Tags: algebra , divisibility

Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$

2010 Contests, 2

Tags: trigonometry , derivative , inequalities , calculus , algebra

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

2023 Belarusian National Olympiad, 9.5

Tags: algebra , polynomial

The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

MOAA Team Rounds, 2022.9

Tags: algebra

Emily has two cups $A$ and $B$, each of which can hold $400$ mL, A initially with $200$ mL of water and $B$ initially with $300$ mL of water. During a round, she chooses the cup with more water (randomly picking if they have the same amount), drinks half of the water in the chosen cup, then pours the remaining half into the other cup and refills the chosen cup to back to half full. If Emily goes for $20$ rounds, how much water does she drink, to the nearest integer?

2018 IMAR Test, 2

Tags: polynomial , minimum value , permutation , algebra

Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum \[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers. [i]Fedor Petrov[/i]

2019 Taiwan TST Round 3, 2

Tags: algebra , polynomial

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

2022 Dutch IMO TST, 2

Tags: algebra

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$

2022 MMATHS, 1

Tags: algebra

Suppose that $ a + b = 20$, $b + c = 22$, and $c + a = 2022$. Compute $\frac{a-b}{c-a}$ .

2022 Princeton University Math Competition, A2 / B4

Tags: algebra

Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$ (The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)

2006 District Olympiad, 3

Tags: algebra , polynomial , superior algebra

Prove that if $A$ is a commutative finite ring with at least two elements and $n$ is a positive integer, then there exists a polynomial of degree $n$ with coefficients in $A$ which does not have any roots in $A$.

LMT Team Rounds 2021+, A22 B23

Tags: algebra

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done? [i]Proposed by Ada Tsui[/i]