problem archive
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Prove that an absolute value \(|\cdot|\) on a field \(k\) is nonarchimedean if and only if \(|n|\leq 1\) for all \(n\in\mathbb{Z}_{>0}\) (view \(n\) a... -
Prove the product formula for \(\mathbb{Q}\): show that \(\prod_{p\leq\infty}|x|_{p}=1\) for all \(x\in\mathbb{Q}^{\times}\). -
Prove Ostrowski's theorem for \(\mathbb{F}_{q}(t)\): every nontrivial absolute value on \(\mathbb{F}_{q}(t)\) is equivalent to \(|\cdot|_{\infty}\) or... -
Prove the product formula for \(\mathbb{F}_{q}(t)\): show that \(\prod_{\pi}|r|_{\pi}=1\) for every \(r\in\mathbb{F}_{q}(t)^{\times}\). -
Show that \(\mathcal{O}_{K}/\mathfrak{p}\simeq\mathbb{F}_{q}\) where \(q=[\mathcal{O}_{K}:\mathfrak{p}]\) is either \(p\) or \(p^{2}\), with \(\mathfr... -
Two numbers \(x\) and \(y\) have a geometric mean of 12 and an arithmetic mean of 12.5. Find \(x^{2}+y^{2}\). (MATHCOUNTS 1992) -
The ace of hearts, the ace of clubs, the ace of diamonds, and the ace of spades are face down on a table. Two different cards are selected at ... -
In a raffle 20 tickets are sold. Two prizes will be given. A student buys 2 tickets. What is the probability that this student wins at least o... -
A teacher with a math class of 20 students randomly pairs the students to take a test. What is the probability that Camilla and Cameron, two students ... -
If there are 3 boys and 4 girls in a group and two are chosen to give a report, what is the probability that one boy and one girl are chosen? (MATHCOU... -
The odds are 7 to 15 against horse Car Naggy winning the third race at Upson Downs. What is the probability that a different horse will win? (MAO 1987... -
The probability that a baseball player gets a hit is 3/10. Find the probability that she gets 2 hits in 4 at bats in her next game. (MAO 1991) -
Two digits between 1 and 9, inclusive, are selected at random. The same digit may be selected twice. What is the probability that their product is a m... -
If five standard fair dice are tossed simultaneously, what is the probability that the outcome has a sum greater than 28? (MATHCOUNTS 1989) -
Eight first-graders, 4 girls and 4 boys, arrange themselves at random around a merry-go-round. What is the probability that boys and girls will be sea... -
A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes.... -
The integers from 1 to 10, inclusive, are partitioned at random into two sets of five elements each. What is the probability that 1 and 2 are in the s... -
Ashley, Bob, Carol, and Doug are rescued from a desert island by a pirate who forces them to play a game. Each of the four, in alphabetical order by f... -
Rocky and Bullwinkle are playing Risk. Rocky rolls one six sided die, while Bullwinkle rolls two of them. What is the probability that Rocky's roll is... -
Find the value of \( \arcsin\left( \frac{1}{2} \right) \)