MathDB

Individual

Part of 2023 CUBRMC

Problems(1)

2023 CUBRMC Individual Round = Cornell University Big Red Math Competition

Source:

3/9/2024
p1. Find the largest 44 digit integer that is divisible by 22 and 55, but not 33.
p2. The diagram below shows the eight vertices of a regular octagon of side length 22. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure 270o270^o. Compute the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png
p3. Consider the numbers formed by writing full copies of 20232023 next to each other, like so: 2023202320232023...2023202320232023... How many copies of 20232023 are next to each other in the smallest multiple of 1111 that can be written in this way?
p4. A positive integer nn with base-1010 representation n=a1a2...akn = a_1a_2 ...a_k is called powerful if the digits aia_i are nonzero for all 1≤i≤k1 \le i \le k and n=a1a1+a2a2+...+akak.n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k . What is the unique four-digit positive integer that is powerful?
p5. Six (6)(6) chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation.
p6. Given that the infinite sum 114+124+134+...\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +... is equal to π490\frac{\pi^4}{90}, compute the value of 114+124+134+...114+134+154+...\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}
p7. Triangle ABCABC is equilateral. There are 33 distinct points, XX, YY , ZZ inside △ABC\vartriangle ABC that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio 1:1:21 : 1 : 2 in some order. Find the ratio of the area of △ABC\vartriangle ABC to that of △XYZ\vartriangle XY Z.
p8. For a fixed prime pp, a finite non-empty set S={s1,...,sk}S = \{s_1,..., s_k\} of integers is pp-admissible if there exists an integer nn for which the product (s1+n)(s2+n)...(sk+n)(s_1 + n)(s_2 + n) ... (s_k + n) is not divisible by pp. For example, {4,6,8}\{4, 6, 8\} is 22-admissible since (4+1)(6+1)(8+1)=315(4+1)(6+1)(8+1) = 315 is not divisible by 22. Find the size of the largest subset of {1,2,...,360}\{1, 2,... , 360\} that is two-,three-, and five-admissible.
p9. Kwu keeps score while repeatedly rolling a fair 66-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a 11, he stops rolling. For example, if the first roll is 11, he gets a score of 11, and if he rolls the sequence (3,4,1)(3, 4, 1), he gets a score of 3+3⋅4+2⋅1=173 + 3 \cdot 4 + 2 \cdot 1 = 17. What is his expected score?
p10. Let {a1,a2,a3,...}\{a_1, a_2, a_3, ...\} be a geometric sequence with a1=4a_1 = 4 and a2023=14a_{2023} = \frac14 . Let f(x)=17(1+x2)f(x) = \frac{1}{7(1+x^2)}. Find f(a1)+f(a2)+...+f(a2023).f(a_1) + f(a_2) + ... + f(a_{2023}).
p11. Let SS be the set of quadratics x2+ax+bx^2 + ax + b, with aa and bb real, that are factors of x14−1x^{14} - 1. Let f(x)f(x) be the sum of the quadratics in SS. Find f(11)f(11).
p12. Find the largest integer 0<n<1000 < n < 100 such that n2+2nn^2 + 2n divides 4(n−1)!+n+44(n- 1)! + n + 4.
p13. Let ω\omega be a unit circle with center OO and radius OQOQ. Suppose PP is a point on the radius OQOQ distinct from QQ such that there exists a unique chord of ω\omega through PP whose midpoint when rotated 120o120^o counterclockwise about QQ lies on ω\omega. Find OPOP.
p14. A sequence of real numbers {ai}\{a_i\} satisfies n⋅a1+(n−1)⋅a2+(n−2)⋅a3+...+2⋅an−1+1⋅an=2023nn \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n for each integer n≥1n \ge 1. Find the value of a2023a_{2023}.
p15. In △ABC\vartriangle ABC, let ∠ABC=90o\angle ABC = 90^o and let II be its incenter. Let line BIBI intersect ACAC at point DD, and let line CICI intersect ABAB at point EE. If ID=IE=1ID = IE = 1, find BIBI.
p16. For a positive integer nn, let SnS_n be the set of permutations of the first nn positive integers. If p=(a1,...,an)∈Snp = (a_1, ..., a_n) \in S_n, then define the bijective function σp:{1,...,n}→{1,...,n}\sigma_p : \{1,..., n\} \to \{1, ..., n\} such that σp(i)=ai\sigma_p (i) = a_i for all integers 1≤i≤n1 \le i \le n. For any two permutations p,q∈Snp, q \in S_n, we say pp and qq are friends if there exists a third permutation r∈Snr \in S_n such that for all integers 1≤i≤n1 \le i \le n, σp(σr(i))=σr(σq(i)).\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)). Find the number of friends, including itself, that the permutation (4,5,6,7,8,9,10,2,3,1)(4, 5, 6, 7, 8, 9, 10, 2, 3, 1) has in S10S_{10}.
PS. You had better use hide for answers.
algebrageometrycombinatoricsnumber theorycubrmc