Olympiad problems

2001 China Team Selection Test, 1

For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of \[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.

1995 Greece National Olympiad, 3

Tags: algebra unsolved , algebra

If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

2002 Austrian-Polish Competition, 4

Tags: inequalities , algebra unsolved , algebra

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

2012 Nordic, 1

Tags: algebra unsolved , algebra

The real numbers $a, b, c$ are such that $a^2 + b^2 = 2c^2$, and also such that $a \ne b, c \ne -a, c \ne -b$. Show that \[\frac{(a+b+2c)(2a^2-b^2-c^2)}{(a-b)(a+c)(b+c)}\] is an integer.

2008 China Girls Math Olympiad, 2

Tags: algebra , polynomial , inequalities , algebra unsolved

Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that \[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0. \]

2011 BMO TST, 1

Tags: analytic geometry , conics , parabola , function , algebra unsolved , algebra

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

1993 China Team Selection Test, 2

Tags: quadratics , function , algebra unsolved , algebra , inequalities

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

1997 Finnish National High School Mathematics Competition, 1

Tags: parameterization , quadratics , logarithms , algebra , polynomial , product of roots , algebra unsolved

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2014 Iran MO (3rd Round), 6

Tags: algebra , polynomial , function , floor function , algebra unsolved

$P$ is a monic polynomial of odd degree greater than one such that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] (a) Prove that there are a finite number of natural numbers in range of $f$. (b) Prove that if $f$ is not constant then the equation $P(x)-x=0$ has at least two real solutions. (c) For each natural $n>1$ prove that there exists a function $f : \mathbb{R} \rightarrow \mathbb{N}$ and a monic polynomial of odd degree greater than one $P$ such that for each $x \in \mathbb{R}$ ,\[f(P(x))=P(f(x))\] and range of $f$ contains exactly $n$ different numbers. Time allowed for this problem was 105 minutes.

2001 Tournament Of Towns, 3

Tags: algebra unsolved , algebra

Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first $10$ customers has made their purchase, that there just enough cheese for the next $10$ customers? If so, how much cheese will be left in the store after the first $10$ customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)

1998 Canada National Olympiad, 2

Tags: algebra unsolved , algebra

Find all real numbers $x$ such that: \[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]

2008 Czech and Slovak Olympiad III A, 1

Tags: algebra unsolved , algebra

Find all pairs of real numbers $(x,y)$ satisfying: \[x+y^2=y^3,\]\[y+x^2=x^3.\]

2000 Bulgaria National Olympiad, 1

Tags: algebra , polynomial , algebra unsolved

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

2010 Contests, 2

Tags: induction , algebra unsolved , algebra

We denote $N_{2010}=\{1,2,\cdots,2010\}$ [b](a)[/b]How many non empty subsets does this set have? [b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products? [b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$. Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.

2011 Macedonia National Olympiad, 4

Tags: function , search , algebra unsolved , algebra

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

2011 Indonesia TST, 1

Tags: algebra unsolved , number theory unsolved , number theory , algebra

Find all real number $x$ which could be represented as $x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$

2004 India IMO Training Camp, 2

Tags: algebra , polynomial , algebra unsolved

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$

2002 Irish Math Olympiad, 3

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that: $ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.

1984 Iran MO (2nd round), 3

Tags: function , calculus , derivative , geometry , geometric transformation , algebra unsolved , algebra

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

2001 Tournament Of Towns, 1

Tags: function , algebra unsolved , algebra

A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)

2004 Vietnam National Olympiad, 1

Tags: trigonometry , limit , algebra unsolved , algebra

The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.

1977 IMO Longlists, 46

Tags: inequalities , function , algebra unsolved , algebra

Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\] Assume that the inequalities\[2^{-1} < g(x, t) < 2\] hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise. Show that\[ 14^{-1} < g(x, t) < 14\] for all real $x$ and positive $t.$

2005 USA Team Selection Test, 4

Tags: algebra , polynomial , algebra unsolved

Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.

2008 Federal Competition For Advanced Students, Part 2, 1

Tags: function , modular arithmetic , algebra unsolved , algebra

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.

2011 IFYM, Sozopol, 1

Tags: algebra , polynomial , floor function , algebra unsolved

Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.