Found problems: 492
2000 National High School Mathematics League, 4
Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$?
$\text{(A)}$ It has no real roots.
$\text{(B)}$ It has two equal real roots.
$\text{(C)}$ It has two different real roots, and their product is positive.
$\text{(D)}$ It has two different real roots, and their product is negative.
2013 Math Prize For Girls Problems, 6
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
2020 CCA Math Bonanza, I13
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$.
[i]2020 CCA Math Bonanza Individual Round #13[/i]
1992 IMO Longlists, 46
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
2002 National High School Mathematics League, 8
Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.
2018 IMO Shortlist, N7
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
2020 Thailand TST, 3
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying
\[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\]
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set
\[X_v=\{x\in\mathbb Z:f(x)=v\}\]
is finite and nonempty.
(a) Prove that there exists such a function $f$ for which there is an $f$-rare integer.
(b) Prove that no such function $f$ can have more than one $f$-rare integer.
[i]Netherlands[/i]
2017 Mathematical Talent Reward Programme, SAQ: P 5
Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$
1978 Putnam, A1
Let $A$ be any set of $20$ distinct integers chosen from the arithmetic progression $1, 4, 7,\ldots,100$. Prove that there must be two distinct integers in $A$ whose sum is $104$.
2014 Cezar Ivănescu, 1
[b]a)[/b] Find the real numbers $ x,y $ such that the set $ \{ x,y \}\cup\left\{ 31/20,29/30,27/40,11/120 \right\} $ contains six elements that can represent an arithmetic progression.
[b]b)[/b] Let be four real numbers in arithmetic progression $ b_1<b_2<b_3<b_4. $ Are there sets $ S $ of $ 6 $ elements that represent an arithmetic progression such that $ \left\{ b_1,b_2,b_3,b_4 \right\}\subset S? $
2015 Estonia Team Selection Test, 1
Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.
2002 Estonia National Olympiad, 1
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.
2016 CCA Math Bonanza, L3.4
Let $S$ be the set of the reciprocals of the first $2016$ positive integers and $T$ the set of all subsets of $S$ that form arithmetic progressions. What is the largest possible number of terms in a member of $T$?
[i]2016 CCA Math Bonanza Lightning #3.4[/i]
2007 Czech and Slovak Olympiad III A, 4
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?
2018 China Team Selection Test, 4
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
1999 AIME Problems, 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
2017 Israel Oral Olympiad, 7
The numbers $1,...,100$ are written on the board. Tzvi wants to colour $N$ numbers in blue, such that any arithmetic progression of length 10 consisting of numbers written on the board will contain blue number. What is the least possible value of $N$?
2012-2013 SDML (High School), 14
A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$.
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
2001 Baltic Way, 2
Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.
1977 Polish MO Finals, 3
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
2000 Estonia National Olympiad, 1
Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence.
Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.
2009 Putnam, B3
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
2006 IberoAmerican Olympiad For University Students, 2
Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation
\[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\]
has at least one real root.
1999 Mexico National Olympiad, 2
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
1973 Kurschak Competition, 1
For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?