Olympiad problems

2004 IberoAmerican, 1

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

1986 Greece Junior Math Olympiad, 4

Tags: algebra

i) If $b=\dfrac{a^2+ \dfrac{1}{a^2}}{a^2-\dfrac{1}{a^2}}$ , express $c=\dfrac{a^4+\dfrac{1}{a^4}}{a^4-\dfrac{1}{a^4}}$ , in terms of $b$. ii) If $k= \frac{x^{n}+\dfrac{1}{x^{n}}}{x^{n}-\dfrac{1}{x^{n}}}$, express $m= \frac{x^{2n}+\dfrac{1}{x^{2n}}}{x^{2n}-\dfrac{1}{x^{2n}}}$ in terms of $k$.

1991 Swedish Mathematical Competition, 2

Tags: inequalities , algebra

$x, y$ are positive reals such that $x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}$. Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$.

2021 BMT, Tie 1

Tags: algebra

Let the sequence $\{a_n\}$ for $n \ge 0$ be defined as $a_0 = c$, and for $n \ge 0$, $$a_n =\frac{2a_{n-1}}{4a^2_{n-1} -1}.$$ Compute the sum of all values of $c$ such that $a_{2020}$ exists but $a_{2021}$ does not exist.

2018 Mathematical Talent Reward Programme, SAQ: P 5

Tags: fibonacci sequence , periodic sequence , remainder , polynomial , algebra

[list=1] [*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime. [/list]

2012 Junior Balkan MO, 4

Tags: algebra , modular arithmetic , greatest common divisor , number theory

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

1995 China Team Selection Test, 3

Tags: polynomial , quadratic , algebra

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

2015 Saudi Arabia IMO TST, 3

Tags: min , max , inequalities , algebra

Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

2003 Romania Team Selection Test, 4

Tags: ratio , floor function , algebra , geometric sequence

Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer. [i]Radu Gologan[/i]

2008 District Olympiad, 2

Tags: number theory , polynomial , algebra

Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.

2020 Bosnia and Herzegovina Junior BMO TST, 4

Tags: algebra

Determine the largest positive integer $n$ such that the following statement holds: If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.

2006 IberoAmerican Olympiad For University Students, 3

Tags: polynomial , algebra , trigonometry , induction

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$. Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.

2008 USA Team Selection Test, 5

Tags: pythagorean theorem , geometry , inequalities , circumcircle , analytic geometry , algebra

Two sequences of integers, $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots$, satisfy the equation \[ (a_n \minus{} a_{n \minus{} 1})(a_n \minus{} a_{n \minus{} 2}) \plus{} (b_n \minus{} b_{n \minus{} 1})(b_n \minus{} b_{n \minus{} 2}) \equal{} 0 \] for each integer $ n$ greater than $ 2$. Prove that there is a positive integer $ k$ such that $ a_k \equal{} a_{k \plus{} 2008}$.

2008 Postal Coaching, 1

Tags: number theory , irrational , fraction , natural , algebra , floor function

For each positive $ x \in \mathbb{R}$, define $ E(x)=\{[nx]: n\in \mathbb{N}\}$ Find all irrational $ \alpha >1$ with the following property: If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.

2024 Azerbaijan JBMO TST, 4

Tags: inequality , algebra

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

1999 Vietnam National Olympiad, 1

Tags: inequalities , algebra , trigonometry

Given are three positive real numbers $ a,b,c$ satisfying $ abc \plus{} a \plus{} c \equal{} b$. Find the max value of the expression: \[ P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.\]

2020 BMT Fall, Tie 2

Tags: algebra , combinatorics

Let $\eta \in [0, 1]$ be a relative measure of material absorbence. $\eta$ values for materials combined together are additive. $\eta$ for a napkin is $10$ times that of a sheet of paper, and a cardboard roll has $\eta = 0.75$. Justin can create a makeshift cup with $\eta = 1$ using $50$ napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with $\eta = 1$?

1998 Austrian-Polish Competition, 1

Tags: inequalities , algebra

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

2018 Moldova Team Selection Test, 6

Tags: three variable inequality , inequalities , algebra

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that $$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$

2011 Croatia Team Selection Test, 1

Tags: modular arithmetic , pigeonhole principle , algebra , induction

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

2008 ITAMO, 3

Tags: function , floor function , algebra , induction

Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions: (i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$; (ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.

2019 Ramnicean Hope, 1

Tags: equation , algebra , monotone

Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/i]

2014 Online Math Open Problems, 12

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

Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

1988 Bulgaria National Olympiad, Problem 1

Tags: algebra

Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$has four real roots.

2024 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra , combinatorics

The $100 \times 100$ table is filled with numbers from $1$ to $10 \ 000$ as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?