Found problems: 25
2009 Stanford Mathematics Tournament, 5
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes
2001 Stanford Mathematics Tournament, 14
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
2001 Stanford Mathematics Tournament, 11
Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an infinite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses first, what is the probability that he will win? Express your answer as a fraction.
2001 Stanford Mathematics Tournament, 9
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.
2001 Stanford Mathematics Tournament, 3
Find the 2000th positive integer that is not the difference between any two integer squares.
2001 Stanford Mathematics Tournament, 5
What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)
2001 Stanford Mathematics Tournament, 13
You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?
1987 IMO Longlists, 2
Suppose we have a pack of $2n$ cards, in the order $1, 2, . . . , 2n$. A perfect shuffle of these cards changes the order to $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$ ; i.e., the cards originally in the first $n$ positions have been moved to the places $2, 4, . . . , 2n$, while the remaining $n$ cards, in their original order, fill the odd positions $1, 3, . . . , 2n - 1.$
Suppose we start with the cards in the above order $1, 2, . . . , 2n$ and then successively apply perfect shuffles.
What conditions on the number $n$ are necessary for the cards eventually to return to their original order? Justify your answer.
[hide="Remark"]
Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.[/hide]
2001 Stanford Mathematics Tournament, 12
A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.
2001 Stanford Mathematics Tournament, 6
Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9.
2006 Stanford Mathematics Tournament, 4
Rice University and Stanford University write questions and corresponding solutions for a high school math tournament. The Rice group writes 10 questions every hour but make a mistake in calculating their solutions 10% of the time. The Stanford group writes 20 problems every hour and makes solution mistakes 20% of the time. Each school works for 10 hours and then sends all problems to Smartie to be checked. However, Smartie isn’t really so smart, and only 75% of the problems she thinks are wrong are actually incorrect. Smartie thinks 20% of questions from Rice have incorrect solutions, and that 10% of questions from Stanford have incorrect solutions. This problem was definitely written and solved correctly. What is the probability that Smartie thinks its solution is wrong?
2001 Stanford Mathematics Tournament, 7
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.
2007 Stanford Mathematics Tournament, 6
Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?
2004 Postal Coaching, 2
(a) Find all triples $(x,y,z)$ of positive integers such that $xy \equiv 2 (\bmod{z})$ , $yz \equiv 2 (\bmod{x})$ and $zx \equiv 2 (\bmod{y} )$
(b) Let $n \geq 1$ be an integer. Give an algoritm to determine all triples $(x,y,z)$ such that '2' in part (a) is replaced by 'n' in all three congruences.
2001 Stanford Mathematics Tournament, 8
Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.
1992 Putnam, B2
For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be the coefficient of $x^{k}$ in the expansion $(1+x+x^{2}+x^{3})^{n}$. Prove that
$Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}$.
[hide="hint"]
Think of $\binom{n}{j}$ as the number of ways you can pick the $x^{2}$ term in the expansion.[/hide]
2001 Stanford Mathematics Tournament, 15
Let $ABC$ be an isosceles triangle with $\angle{ABC} = \angle{ACB} = 80^\circ$. Let $D$ be a point on $AB$ such that $\angle{DCB} = 60^\circ$ and $E$ be a point on $AC$ such that $\angle{ABE} = 30^\circ$. Find $\angle{CDE}$ in degrees.
2005 Putnam, A3
Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$
2001 Stanford Mathematics Tournament, 10
You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$.
2001 Stanford Mathematics Tournament, 1
$ABCD$ is a square with sides of unit length. Points $E$ and $F$ are taken on sides $AB$ and $AD$ respectively so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. What is this maximum area?
2013 NIMO Problems, 3
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
2001 Stanford Mathematics Tournament, 2
How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?
2006 Putnam, B1
Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.
2007 ITest, 26
Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?
2001 Stanford Mathematics Tournament, 4
For what values of $a$ does the system of equations
\[x^2 = y^2,(x-a)^2 +y^2 = 1\]have exactly 2 solutions?