Found problems: 20
2022 Czech and Slovak Olympiad III A, 2
We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair?
(A palindrome is an integer that reads the same forward and backward.)
[i](David Hruska)[/i]
2024 Middle European Mathematical Olympiad, 4
A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers
$1 \le i \le r$.
Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by
$a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite
sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence
$a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such
that $a_i=a_{i+T}$ for every positive integer $i$.
1991 Mexico National Olympiad, 2
A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.
2022 Denmark MO - Mohr Contest, 2
A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.
2017 Romanian Master of Mathematics Shortlist, N2
Let $x, y$ and $k$ be three positive integers. Prove that there exist a positive integer $N$ and a set of $k + 1$ positive integers $\{b_0,b_1, b_2, ... ,b_k\}$, such that, for every $i = 0, 1, ... , k$ , the $b_i$-ary expansion of $N$ is a $3$-digit palindrome, and the $b_0$-ary expansion is exactly $\overline{\mbox{xyx}}$.
proposed by Bojan Basic, Serbia
2024 Philippine Math Olympiad, P5
Find the largest positive integer $k$ so that any binary string of length $2024$ contains a palindromic substring of length at least $k$.
2019 Saint Petersburg Mathematical Olympiad, 1
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.
1998 VJIMC, Problem 2
Decide whether there is a member in the arithmetic sequence $\{a_n\}_{n=1}^\infty$ whose first member is $a_1=1998$ and the common difference $d=131$ which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g., $7$, $33$, $433334$, $2135312$ and so on).
2014 JHMMC 7 Contest, 5
A palindrome is a word that reads the same backwards as forwards, such as “eye”, “race car”, and “qwertyytrewq”.
How many letters are in the smallest palindrome containing the letters b, o, g, t, r, and o, not necessarily in that order
and not necessarily adjacent?
2022/2023 Tournament of Towns, P2
Does there exist a natural number that can be represented as the product of two numeric palindromes in more than $100{}$ ways?
2006 Dutch Mathematical Olympiad, 1
A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap.
How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions:
- each letter may appear no more than twice in each palindrome,
- the length of each palindrome is at least $3$ letters.
(Any possible combination of letters is considered a word.)
2014 Danube Mathematical Competition, 2
We call [i]word [/i] a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ .
A [i]word [/i] $\overline {l_1l_2...l_n}, n\ge 1$ is called [i]palindrome [/i] if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$.
Consider a [i]word [/i] $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$.
Prove that there are at least $806$ [i]palindrome [/i] [i]words [/i] to ''stick" together to get word $X$.
2018 Regional Olympiad of Mexico Center Zone, 1
Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values of $N-M$.
2009 Dutch Mathematical Olympiad, 1
In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satis es the following two conditions:
- the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around);
- the integer is a product of two positive integers, of which the fi rst, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$.
For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$, and $20502$ itself is a palindrome.
Determine all palindromic products of $5$ digits.
2005 Paraguay Mathematical Olympiad, 3
The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .
2016 Balkan MO Shortlist, C1
Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.
2024 Czech-Polish-Slovak Junior Match, 5
Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?
2009 Bundeswettbewerb Mathematik, 4
A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.
1983 AIME Problems, 10
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
2015 Caucasus Mathematical Olympiad, 5
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?