Olympiad problems

2002 Korea - Final Round, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.

2010 Postal Coaching, 3

Tags: function , algebra

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\boxed{1} \ f(1) = 1$ $\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$

1990 IMO Longlists, 14

Tags: algebra , transformation , translation , invariant

We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.

2023 Malaysian IMO Training Camp, 3

Tags: algebra

A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$ [i]Proposed by Ivan Chan Kai Chin[/i]

2012 Dutch IMO TST, 5

Tags: functional equation , algebra

Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.

2009 AMC 8, 23

Tags: quadratic , algebra , quadratic formula

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? $ \textbf{(A)}\ 26 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 34$

1969 Leningrad Math Olympiad, 7.4*

Tags: algebra

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

2024 IFYM, Sozopol, 1

Tags: functional equation , algebra

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+} $ such that: \[ f(x^2 + y) = xf(x) + \frac{f(y^2)}{y} \] for any positive real numbers $ x $ and $ y $.

2014 IFYM, Sozopol, 6

Tags: algebra , coordinates , circles

Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?

1997 Argentina National Olympiad, 3

Tags: algebra , sum , inequalities

Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$

2002 China Girls Math Olympiad, 6

Tags: induction , algebra , number theory , equation

Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

2017 Baltic Way, 3

Tags: fibonacci , additive number theory , extremal combinatorics , algebra

Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$? (Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)

2023 Indonesia TST, 1

Tags: algebra , sequence

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

2018 Estonia Team Selection Test, 3

Tags: sum , algebra , inequalities , max , min

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 6

Tags: algebra , functional equation

Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all the functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$, \[ f(x)f(y) = f(y)f(xf(y)) + \frac{1}{xy}. \]

LMT Team Rounds 2010-20, B16

Tags: algebra

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

2023 Indonesia TST, A

Tags: polynomial , integer polynomial , algebra

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

PEN Q Problems, 13

Tags: algebra , algorithm , polynomial

On Christmas Eve, 1983, Dean Jixon, the famous seer who had made startling predictions of the events of the preceding year that the volcanic and seismic activities of $1980$ and $1981$ were connected with mathematics. The diminishing of this geological activity depended upon the existence of an elementary proof of the irreducibility of the polynomial \[P(x)=x^{1981}+x^{1980}+12x^{2}+24x+1983.\] Is there such a proof?

2017-IMOC, A6

Tags: inequalities , algebra

Show that for all positive reals $a,b,c$ with $a+b+c=3$, $$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$

1976 Chisinau City MO, 124

Tags: algebra , system of equations

Find $3$ numbers, each of which is equal to the square of the difference of the other two.

2017 BMT Spring, 4

Tags: algebra

Find the value of $\frac12+\frac{4}{2^2} +\frac{9}{2^3} +\frac{16}{2^4} + ...$

2016 India IMO Training Camp, 2

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2016 Hong Kong TST, 2

Tags: algebra

Determine all positive integers $n$ for which there exist pairwise distinct positive real numbers $a_1, a_2, \cdots, a_n$ satisfying $\displaystyle \left\{a_i+\frac{(-1)^i}{a_i}\mid 1\leq i \leq n\right\}=\{a_i\mid 1\leq i \leq n\}$

2020 LMT Fall, B29

Tags: algebra

Alicia bought some number of disposable masks, of which she uses one per day. After she uses each of her masks, she throws out half of them (rounding up if necessary) and reuses each of the remaining masks, repeating this process until she runs out of masks. If her masks lasted her $222$ days, how many masks did she start out with?

MathLinks Contest 5th, 3.1

Tags: algebra

Let $\{x_n\}_n$ be a sequence of positive rational numbers, such that $x_1$ is a positive integer, and for all positive integers $n$. $x_n = \frac{2(n - 1)}{n} x_{n-1}$, if $x_{n_1} \le 1$ $x_n = \frac{(n - 1)x_{n-1} - 1}{n}$ , if $x_{n_1} > 1$. Prove that there exists a constant subsequence of $\{x_n\}_n$.