Found problems: 38
1980 IMO Shortlist, 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 $s \geq 0, 0 < q < p$.
2014 AMC 12/AHSME, 23
The number $2017$ is prime. Let $S=\sum_{k=0}^{62}\binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
$\textbf{(A) }32\qquad
\textbf{(B) }684\qquad
\textbf{(C) }1024\qquad
\textbf{(D) }1576\qquad
\textbf{(E) }2016\qquad$
1993 Tournament Of Towns, (371) 3
Each number in the second, third, and further rows of the following triangle:
[img]https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png[/img]
is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle).
(GW Leibnitz,)
2008 ITest, 77
With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.
There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\\\dbinom{16}5&=4368.\end{align*}
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test:
\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]
Wendy works the solution out correctly. What is her answer?
2008 China Northern MO, 2
The given triangular number table is as follows:
[img]https://cdn.artofproblemsolving.com/attachments/a/0/123b7511850047f3cc51494f107703f2757085.png[/img]
Among them, the numbers in the first row are $1, 2, 3, ..., 98, 99, 100$. Starting from the second row, each number is equal to the sum of the left and right numbers in the row above it. Find the value of $M$.
2000 Estonia National Olympiad, 2
The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table).
We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another.
[img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]
2013 Bundeswettbewerb Mathematik, 4
Consider the Pascal's triangle in the figure where the binomial coefficients are arranged in the usual manner. Select any binomial coefficient from anywhere except the right edge of the triangle and labet it $C$. To the right of $C$, in the horizontal line, there are $t$ numbers, we denote them as $a_1,a_2,\cdots,a_t$, where $a_t = 1$ is the last number of the series. Consider the line parallel to the left edge of the triangle containing $C$, there will only be $t$ numbers diagonally above $C$ in that line. We successively name them as $b_1,b_2,\cdots,b_t$, where $b_t = 1$. Show that
\[b_ta_1-b_{t-1}a_2+b_{t-2}a_3-\cdots+(-1)^{t-1}b_1a_t = 1\].
For example, Suppose you choose $\binom41 = 4$ (see figure), then $t = 3$, $a_1 = 6, a_2 = 4, a_3 = 1$ and $b_1 = 3, b_2 = 2, b_3 = 1$.
\[\begin{array}{ccccccccccc} & & & & & 1 & & & & & \\
& & & & 1 & & \underset{b_3}{1} & & & & \\
& & & 1 & & \underset{b_2}{2} & & 1 & & & \\
& & 1 & & \underset{b_1}{3} & & 3 & & 1 & & \\
& 1 & & \boxed{4} & & \underset{a_1}{6} & & \underset{a_2}{4} & & \underset{a_3}{1} & \\
\ldots & & \ldots & & \ldots & & \ldots & & \ldots & & \ldots \\
\end{array}\]
2007 AIME Problems, 13
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
[asy]
defaultpen(linewidth(0.7));
path p=origin--(1,0)--(1,1)--(0,1)--cycle;
int i,j;
for(i=0; i<12; i=i+1) {
for(j=0; j<11-i; j=j+1) {
draw(shift(i/2+j,i)*p);
}}[/asy]
1990 IMO Longlists, 44
Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.
1976 AMC 12/AHSME, 25
For a sequence $u_1,u_2\dots,$ define $\Delta^1(u_n)=u_{n+1}-u_n$ and, for all integer $k>1$, $\Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n))$. If $u_n=n^3+n$, then $\Delta^k(u_n)=0$ for all $n$
$\textbf{(A) }\text{if }k=1\qquad$
$\textbf{(B) }\text{if }k=2,\text{ but not if }k=1\qquad$
$\textbf{(C) }\text{if }k=3,\text{ but not if }k=2\qquad$
$\textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad$
$\textbf{(E) }\text{for no value of }k$
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?
2011 BAMO, 5
Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$?
Recall that Pascal’s triangle is the pattern of numbers that begins as follows
[img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img]
where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$.
Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.
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$.