Found problems: 15925
2007 IMS, 6
Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.
2013 Putnam, 5
Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$
[Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]
1993 IMO Shortlist, 7
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
2009 HMNT, 1
Paul starts with the number $19$. In one step, he can add $1$ to his number, divide his number by $2$, or divide his number by $3$. What is the minimum number of steps Paul needs to get to $1$?
1998 Estonia National Olympiad, 3
On a closed track, clockwise, there are five boxes $A, B, C, D$ and $E$, and the length of the track section between boxes $A$ and $B$ is $1$ km, between $B$ and $C$ - $5$ km, between $C$ and $D$ - $2$ km, between $D$ and $E$ - $10$ km, and between $E$ and $A$ - $3$ km. On the track, they drive in a clockwise direction, the race always begins and ends in the box. What box did you start from if the length of the race was exactly $1998$ km?
2010 Dutch IMO TST, 2
Let $A$ and $B$ be positive integers. De fine the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2025 Portugal MO, 1
Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?
2018 Azerbaijan Junior NMO, 2
$x^{11}+x^7+x^3=1$. $$x^{\alpha}=x^4+x^3-1.\hspace{4mm} \alpha=?$$
1982 IMO Longlists, 34
Let $M$ be the set of all functions $f$ with the following properties:
[b](i)[/b] $f$ is defined for all real numbers and takes only real values.
[b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$
[b](iii)[/b] $f(0) \neq 0.$
Determine all functions $f \in M$ such that
[b](a)[/b] $f(1)=\frac 52$,
[b](b)[/b] $f(1)= \sqrt 3$.
2019 Thailand TSTST, 3
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying
$\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$,
$\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and
$\text{(iii)}$ $f(1)-f(0)$ is even.
2010 Ukraine Team Selection Test, 12
Is there a positive integer $n$ for which the following holds:
for an arbitrary rational $r$ there exists an integer $b$ and non-zero integers $a _1, a_2, ..., a_n$ such that $r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}$ ?
1996 China Team Selection Test, 2
Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that
\[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\]
Define
\begin{align*}
A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\
B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\
W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2.
\end{align*}
Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.
2019 South Africa National Olympiad, 5
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
$$
f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3
$$
for all integers $a, b, c$.
2022-23 IOQM India, 4
Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$
2019 Purple Comet Problems, 3
The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1986 Austrian-Polish Competition, 2
The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$
2009 Cuba MO, 7
Let $x_1, x_2, ..., x_n$ be positive reals. Prove that
$$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$
In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.
2020 Jozsef Wildt International Math Competition, W47
Let $x,y,z>0$ such that
$$(x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10}$$
Compute
$$\left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right]$$
where $[.]$ represents the integer part.
[i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]
1969 IMO Longlists, 67
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.
2018 Turkey Team Selection Test, 6
$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either
$$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$
or
$$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$
Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?
2017 Iran MO (3rd round), 3
Let $a,b$ and $c$ be positive real numbers. Prove that
$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
2019 Israel National Olympiad, 1
In kindergarden, there are 32 children and three classes: Judo, Agriculture, and Math. Every child is in exactly one class and every class has at least one participant.
One day, the teacher gathered 6 children to clean up the classroom. The teacher counted and found that exactly 1/2 of the Judo students, 1/4 of the Agriculture students and 1/8 of the Math students are cleaning.
How many children are in each class?
2014 China Girls Math Olympiad, 7
Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$
2020 Abels Math Contest (Norwegian MO) Final, 3
Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .
1978 IMO Longlists, 6
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.