Found problems: 2008
2014 AMC 10, 24
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
2012 Brazil National Olympiad, 3
Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.
2014 Taiwan TST Round 3, 1
Consider a $6 \times 6$ grid. Define a [i]diagonal[/i] to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals.
Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.
the 14th XMO, P4
In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.
2005 International Zhautykov Olympiad, 1
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
2000 IberoAmerican, 1
A regular polygon of $ n$ sides ($ n\geq3$) has its vertex numbered from 1 to $ n$. One draws all the diagonals of the polygon. Show that if $ n$ is odd, it is possible to assign to each side and to each diagonal an integer number between 1 and $ n$, such that the next two conditions are simultaneously satisfied:
(a) The number assigned to each side or diagonal is different to the number assigned to any of the vertices that is endpoint of it.
(b) For each vertex, all the sides and diagonals that have it as an endpoint, have different number assigned.
PEN O Problems, 6
Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.
2023 Serbia Team Selection Test, P5
For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\]
Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.
1987 India National Olympiad, 3
Let $ T$ be the set of all triplets $ (a,b,c)$ of integers such that $ 1 \leq a < b < c \leq 6$ For each triplet $ (a,b,c)$ in $ T$, take number $ a\cdot b \cdot c$. Add all these numbers corresponding to all the triplets in $ T$. Prove that the answer is divisible by 7.
2003 Italy TST, 1
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
2010 Contests, 1
a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct.
$\tab$ $\tab$ $ABC$
$\tab$ $\tab$ $DEF$
[u]$+GHI$[/u]
$\tab$ $\tab$ $\tab$ $J J J$
Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$.
b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).
PEN A Problems, 43
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.
2001 IMO Shortlist, 6
For a positive integer $n$ define a sequence of zeros and ones to be [i]balanced[/i] if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are [i]neighbors[/i] if you can move one of the $2n$ symbols of $a$ to another position to form $b$. For instance, when $n = 4$, the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$.
2006 Baltic Way, 18
For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.
2000 Belarus Team Selection Test, 8.2
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
2014 ELMO Shortlist, 1
Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$?
[i]Proposed by Jesse Zhang[/i]
1980 IMO Shortlist, 6
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
2014 South East Mathematical Olympiad, 1
Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$
2002 National Olympiad First Round, 30
How many integers $0 \leq x < 125$ are there such that $x^3 - 2x + 6 \equiv 0 \pmod {125}$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$
2012 Gulf Math Olympiad, 4
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.
[list](i) Prove that all the discs that he cuts have integral areas.
(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]
2012 AMC 12/AHSME, 9
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
1992 IMO Longlists, 14
Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and
\[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\]
where $a_{n+j} = a_j$. Prove that $n \neq 1992.$
2008 ITest, 18
Find the number of lattice points that the line $19x+20y=1909$ passes through in Quadrant I.
2014 PUMaC Number Theory B, 6
Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.
2008 Hong Kong TST, 3
Show that the equation $ y^{37} \equal{} x^3 \plus{}11\pmod p$ is solvable for every prime $ p$, where $ p\le 100$.