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Letter Dice Puzzle
Jorge has three four-sided dice with a letter on each side. He rolls all three dice 8 times and tries to write a word using the letters he has rolled. On his sheet are now: DAL, DOR, GAT, IER, ONS, RIT, RED, VEN. Which four letters were on each of the three dice?
💡 View Hint
Each word uses exactly one letter from each die. Analyze letter co-occurrences: D appears with A, O, G, I, R, E, V. A appears with D, L, T, R, E, N. Find letters that never appear together (must be on different dice). Build the dice by finding a 3-coloring of the letter graph.
Special Divisor Property
We call a divisor d of a positive integer N proper if d ≠ 1 and d ≠ N. Thus 2 and 3 are proper divisors of 6, and prime numbers never have proper divisors. Determine all positive integers N with at least two proper divisors and with the property that if d > e are two different proper divisors of N, then d−e is also a (not necessarily proper) divisor of N.
💡 View Hint
Consider small cases first. If N = p², the only proper divisor is p, so the condition is vacuously satisfied... wait, N needs at least two proper divisors. Try N = p³ (proper divisors: p, p²; difference p²-p = p(p-1), must divide p³, so p-1 divides p², giving p=2, N=8). Try N = pq (proper divisors: p, q; difference |p-q| must divide pq). Try N = p²q, etc.
Digit Sum of Page Numbers
Bas has a mathematics book with pages numbered from 1 to 549. He adds up all the digits that appear in the page numbers. What is the result?
💡 View Hint
Count digit occurrences. For pages 1-9: each digit 1-9 appears once. For 10-99: count tens and units separately. For 100-549: count hundreds, tens, and units. Alternatively, use the formula: sum of digits from 1 to n.
Special Year Numbers
It holds that (20+25)² = 2025. There are two more four-digit year numbers ABCD satisfying the equation (AB+CD)² = ABCD. Which year numbers have this very special property?
💡 View Hint
Let AB = 10a + b and CD = 10c + d, or more simply let the first two digits be m and the last two be n. Then (m+n)² = 100m + n. So (m+n)² - (m+n) = 99m, giving (m+n)(m+n-1) = 99m. Let s = m+n, then s(s-1) = 99m and n = s - m = s - s(s-1)/99.
Divisibility by 11
We consider numbers of the form 100...001. For which numbers of zeros is this number divisible by 11?
💡 View Hint
A number is divisible by 11 if the alternating sum of its digits is divisible by 11. For 100...001, the digits are 1, then k zeros, then 1. Use the divisibility rule for 11: 10 ≡ -1 (mod 11), so 10^(k+1) + 1 ≡ (-1)^(k+1) + 1 (mod 11). This is 0 when k+1 is odd, i.e., k is even.
The Machine Function
Bas has a machine where you can insert a card with a positive integer n ≥ 1, after which the machine returns a card with a positive integer on it. If Bas inserts the same card into the machine twice, the same card always comes out. We denote the number on this new card as M(n). The machine satisfies the following rules for all positive integers n and m: If n > m, then M(n) > M(m). It holds that M(M(n)) = 3n. Bas inserts the number 2025 into the machine. What is the result?
💡 View Hint
Write n in base 3. The function M acts on the base-3 representation by shifting digits. If n = (a_k a_{k-1} ... a_0)_3, then M(n) corresponds to a specific digit manipulation. Find the pattern by computing small values.
Self-Referential Cubic
How many triples of real numbers (a, b, c) exist such that the equation x³ + ax² + bx + c = 0 has exactly a, b, and c as its solutions?
💡 View Hint
By Vieta's formulas: a+b+c = -a, ab+bc+ca = b, abc = -c. Analyze cases based on whether c = 0 or c ≠ 0.
Compositions of 10
We can write the number 3 in exactly three ways as a sum of (at least two) positive integers, where the order matters: 3 = 1+2 = 2+1 = 1+1+1. In how many ways can we do this for the number 10?
💡 View Hint
The number of compositions of n into k parts is C(n-1, k-1). Sum over k from 2 to n, or use the recurrence: each composition of n can be formed by adding 1 to the first part of a composition of n-1, or by prepending 1 to a composition of n-1.
Concorde Flight Time
The Concorde was the fastest passenger aircraft ever built. If you departed London at 12:00 noon on a flight to New York, a 6000 km flight, you would arrive there at 11:00 in the morning. This was possible due to the 5-hour time difference between the two cities. The other way around, the Concorde was even faster, because the jet stream would accelerate the aircraft by 250 km/h instead of slowing it down by the same amount. If you departed New York at 12:00 noon with the Concorde to London, what time would you arrive there?
💡 View Hint
Let v be the Concorde's speed without wind. London to New York: distance = 6000 km, time = 11:00 - 12:00 + 5h = 4h (accounting for time zone). So v = 6000/4 = 1500 km/h. New York to London: speed = v + 250 = 1750 km/h (with jet stream). Time = 6000/1750 hours. Add 5h time difference.
Mersenne Prime and Difference of Squares
Since last month, p = 2^136279841 − 1 is the largest known prime number. We consider solutions of the equation x² + p = y², where x and y are positive integers. What are all possible values of x + y?
💡 View Hint
Rewrite as y² - x² = p, so (y-x)(y+x) = p. Since p is prime, the only factorizations are 1·p.
Nested Polynomial and Cosine
Let α = 2cos(π/11). Show that α is a solution of the equation ((((x²−2)²−2)²−2)²−2)²−2 = −x. What are the other solutions of this equation? Prove your answer!
💡 View Hint
Use the identity 2cos(θ)=e^(iθ)+e^(-iθ) and the fact that the 11th roots of unity satisfy a cyclotomic polynomial.
Prime with Trailing Ones
Let n be a positive integer and suppose that the number 7111...111, consisting of a 7 followed by n ones, is prime. Prove that n−1 is divisible by 3.
💡 View Hint
Write the number as 7·10ⁿ + (10ⁿ - 1)/9. Consider the number modulo 3 and use properties of repunits.
Minimum Digit Sum
From a set of 100 different positive integers less than 1000, we add up all their digits. What is the smallest possible result?
💡 View Hint
To minimize the sum of all digits, use numbers with as many zeros as possible. Consider numbers like 100, 200, ..., 900 and fill the rest with single-digit numbers.
Integer Roots Cubic
Determine whether there exists a function f of the form x³ + ax² + bx + c with a, b, c integers satisfying: the solutions of f(x) = 0 are three distinct integers; the solutions of f'(x) = 0 are two distinct integers.
💡 View Hint
Use Vieta's formulas. If f(x)=(x-r)(x-s)(x-t), then f'(x) has roots at the critical points. The derivative roots being integers imposes strong constraints on r, s, t.