Found problems: 38
1947 Moscow Mathematical Olympiad, 139
In the numerical triangle
$................1..............$
$...........1 ...1 ...1.........$
$......1... 2... 3 ... 2 ... 1....$
$.1...3...6...7...6...3...1$
$...............................$
each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
2005 iTest, 3
Find the probability that any given row in Pascal’s Triangle contains a perfect square.
[i] (.1 point)[/i]
1992 AIME Problems, 4
In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.
\[\begin{array}{c@{\hspace{8em}}
c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{4pt}}c@{\hspace{2pt}}
c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{6pt}}
c@{\hspace{6pt}}c@{\hspace{6pt}}c} \vspace{4pt}
\text{Row 0: } & & & & & & & 1 & & & & & & \\\vspace{4pt}
\text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\vspace{4pt}
\text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\vspace{4pt}
\text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt}
\text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt}
\text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\vspace{4pt}
\text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1
\end{array}\]
In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
2000 Mexico National Olympiad, 2
A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row?
1 2 3 4 5
3 5 7 9
8 12 16
20 28
4
1993 Spain Mathematical Olympiad, 2
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.
$0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993$
$\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$
$\,\,\,4 \,8 \,12\, .......... \,\,\,7968$
·······································
Prove that the bottom number is a multiple of $1993$.
2006 Italy TST, 1
Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[/i] if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?
1977 AMC 12/AHSME, 10
If $(3x-1)^7 = a_7x^7 + a_6x^6 + \cdots + a_0$, then $a_7 + a_6 + \cdots + a_0$ equals
\[ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 64 \qquad \text{(D)}\ -64 \qquad \text{(E)}\ 128 \]
1984 Kurschak Competition, 1
Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd?
[img]https://cdn.artofproblemsolving.com/attachments/8/a/4dc4a815d8b002c9f36a6da7ad6e1c11a848e9.png[/img]
1971 AMC 12/AHSME, 24
[asy]
label("$1$",(0,0),S);
label("$1$",(-1,-1),S);
label("$1$",(-2,-2),S);
label("$1$",(-3,-3),S);
label("$1$",(-4,-4),S);
label("$1$",(1,-1),S);
label("$1$",(2,-2),S);
label("$1$",(3,-3),S);
label("$1$",(4,-4),S);
label("$2$",(0,-2),S);
label("$3$",(-1,-3),S);
label("$3$",(1,-3),S);
label("$4$",(-2,-4),S);
label("$4$",(2,-4),S);
label("$6$",(0,-4),S);
label("etc.",(0,-5),S);
//Credit to chezbgone2 for the diagram[/asy]
Pascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\text{th}$ number in any row when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers in the immediately preceding row. The quotient of the number of numbers in the first $n$ rows which are not $1$'s and the number of $1$'s is
$\textbf{(A) }\dfrac{n^2-n}{2n-1}\qquad\textbf{(B) }\dfrac{n^2-n}{4n-2}\qquad\textbf{(C) }\dfrac{n^2-2n}{2n-1}\qquad\textbf{(D) }\dfrac{n^2-3n+2}{4n-2}\qquad \textbf{(E) }\text{None of these}$
2017 Harvard-MIT Mathematics Tournament, 7
Let $p$ be a prime. A [i]complete residue class modulo $p$[/i] is a set containing at least one element equivalent to $k \pmod{p}$ for all $k$.
(a) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.
(b) Show that there exists an $n \le p^2$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.
2013 Harvard-MIT Mathematics Tournament, 11
Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)
1980 IMO, 22
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.
2019 Saudi Arabia Pre-TST + Training Tests, 1.2
Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.
2008 ITest, 78
Feeling excited over her successful explorations into Pascal's Triangle, Wendy formulates a second problem to use during a future Jupiter Falls High School Math Meet:
\[\text{How many of the first 2010 rows of Pascal's Triangle (Rows 0 through 2009)} \ \text{have exactly 256 odd entries?}\]
What is the solution to Wendy's second problem?
1985 All Soviet Union Mathematical Olympiad, 401
In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$?
j
h i
e f g
a b c d
2011 ELMO Shortlist, 3
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.
(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)
[i]Linus Hamilton.[/i]
2006 AMC 12/AHSME, 23
Given a finite sequence $ S \equal{} (a_1,a_2,\ldots,a_n)$ of $ n$ real numbers, let $ A(S)$ be the sequence
\[ \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right)
\]of $ n \minus{} 1$ real numbers. Define $ A^1(S) \equal{} A(S)$ and, for each integer $ m$, $ 2\le m\le n \minus{} 1$, define $ A^m(S) \equal{} A(A^{m \minus{} 1}(S)).$ Suppose $ x > 0$, and let $ S \equal{} (1,x,x^2,\ldots,x^{100})$. If $ A^{100}(S) \equal{} (1/2^{50})$, then what is $ x$?
$ \textbf{(A) } 1 \minus{} \frac {\sqrt {2}}2\qquad \textbf{(B) } \sqrt {2} \minus{} 1\qquad \textbf{(C) } \frac 12\qquad \textbf{(D) } 2 \minus{} \sqrt {2}\qquad \textbf{(E) } \frac {\sqrt {2}}2$
2006 QEDMO 3rd, 5
Find all positive integers $n$ such that there are $\infty$ many lines of Pascal's triangle that have entries coprime to $n$ only. In other words: such that there are $\infty$ many $k$ with the property that the numbers $\binom{k}{0},\binom{k}{1},\binom{k}{2},...,\binom{k}{k}$ are all coprime to $n$.
1980 IMO Longlists, 9
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.
2011 ELMO Shortlist, 3
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.
(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)
[i]Linus Hamilton.[/i]
2009 Indonesia TST, 2
Consider the following array:
\[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots
\] Find the 5-th number on the $ n$-th row with $ n>5$.
2013 Online Math Open Problems, 27
Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads:
(i) If the blackboard is empty, Ben writes $n$ on the blackboard.
(ii) If the blackboard is not empty, let $m$ denote the largest number on the blackboard. If $m^2+2n^2$ is divisible by $3$, Ben erases $m$ from the blackboard; otherwise, he writes the number $n$.
No action is taken when the coin flips tails. If probability that the blackboard is empty after all $2013$ flips is $\frac{2u+1}{2^k(2v+1)}$, where $u$, $v$, and $k$ are nonnegative integers, compute $k$.
[i]Proposed by Evan Chen[/i]
2006 South africa National Olympiad, 5
Find the number of subsets $X$ of $\{1,2,\dots,10\}$ such that $X$ contains at least two elements and such that no two elements of $X$ differ by $1$.
2011 ELMO Problems, 2
Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.
(Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.)
[i]Linus Hamilton.[/i]
1988 AIME Problems, 15
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9.
While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonder which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)