Found problems: 15
2023 CUBRMC, 1
Let $x, y, z$ be positive real numbers. Prove that
$$\sqrt{(z + x)(z + y)} - z \ge \sqrt{xy}.$$
2023 CUBRMC, Individual
[b]p1.[/b] Find the largest $4$ digit integer that is divisible by $2$ and $5$, but not $3$.
[b]p2.[/b] The diagram below shows the eight vertices of a regular octagon of side length $2$. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure $270^o$. Compute the area of the shaded region.
[center][img]https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png[/img][/center]
[b]p3.[/b] Consider the numbers formed by writing full copies of $2023$ next to each other, like so: $$2023202320232023...$$
How many copies of $2023$ are next to each other in the smallest multiple of $11$ that can be written in this way?
[b]p4.[/b] A positive integer $n$ with base-$10$ representation $n = a_1a_2 ...a_k$ is called [i]powerful [/i] if the digits $a_i$ are nonzero for all $1 \le i \le k$ and
$$n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .$$
What is the unique four-digit positive integer that is [i]powerful[/i]?
[b]p5.[/b] Six $(6)$ 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.
[b]p6.[/b] Given that the infinite sum $\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...$ is equal to $\frac{\pi^4}{90}$, compute the value of
$$\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}$$
[b]p7.[/b] Triangle $ABC$ is equilateral. There are $3$ distinct points, $X$, $Y$ , $Z$ inside $\vartriangle ABC$ that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio $1 : 1 : 2$ in some order. Find the ratio of the area of $\vartriangle ABC$ to that of $\vartriangle XY Z$.
[b]p8.[/b] For a fixed prime $p$, a finite non-empty set $S = \{s_1,..., s_k\}$ of integers is $p$-[i]admissible [/i] if there exists an integer $n$ for which the product $$(s_1 + n)(s_2 + n) ... (s_k + n)$$ is not divisible by $p$. For example, $\{4, 6, 8\}$ is $2$-[i]admissible[/i] since $(4+1)(6+1)(8+1) = 315$ is not divisible by $2$. Find the size of the largest subset of $\{1, 2,... , 360\}$ that is two-,three-, and five-[i]admissible[/i].
[b]p9.[/b] Kwu keeps score while repeatedly rolling a fair $6$-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 $1$, he stops rolling. For example, if the first roll is $1$, he gets a score of $1$, and if he rolls the sequence $(3, 4, 1)$, he gets a score of $3 + 3 \cdot 4 + 2 \cdot 1 = 17$. What is his expected score?
[b]p10.[/b] Let $\{a_1, a_2, a_3, ...\}$ be a geometric sequence with $a_1 = 4$ and $a_{2023} = \frac14$ . Let $f(x) = \frac{1}{7(1+x^2)}$. Find $$f(a_1) + f(a_2) + ... + f(a_{2023}).$$
[b]p11.[/b] Let $S$ be the set of quadratics $x^2 + ax + b$, with $a$ and $b$ real, that are factors of $x^{14} - 1$. Let $f(x)$ be the sum of the quadratics in $S$. Find $f(11)$.
[b]p12.[/b] Find the largest integer $0 < n < 100$ such that $n^2 + 2n$ divides $4(n- 1)! + n + 4$.
[b]p13.[/b] Let $\omega$ be a unit circle with center $O$ and radius $OQ$. Suppose $P$ is a point on the radius $OQ$ distinct from $Q$ such that there exists a unique chord of $\omega$ through $P$ whose midpoint when rotated $120^o$ counterclockwise about $Q$ lies on $\omega$. Find $OP$.
[b]p14.[/b] A sequence of real numbers $\{a_i\}$ satisfies
$$n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n$$
for each integer $n \ge 1$. Find the value of $a_{2023}$.
[b]p15.[/b] In $\vartriangle ABC$, let $\angle ABC = 90^o$ and let $I$ be its incenter. Let line $BI$ intersect $AC$ at point $D$, and let line $CI$ intersect $AB$ at point $E$. If $ID = IE = 1$, find $BI$.
[b]p16.[/b] For a positive integer $n$, let $S_n$ be the set of permutations of the first $n$ positive integers. If $p = (a_1, ..., a_n) \in S_n$, then define the bijective function $\sigma_p : \{1,..., n\} \to \{1, ..., n\}$ such that $\sigma_p (i) = a_i$ for all integers $1 \le i \le n$.
For any two permutations $p, q \in S_n$, we say $p$ and $q$ are friends if there exists a third permutation $r \in S_n$ such that for all integers $1 \le i \le n$, $$\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).$$
Find the number of friends, including itself, that the permutation $(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)$ has in $S_{10}$.
PS. You had better use hide for answers.
2023 CUBRMC, 10
Let triangle $ABC$ have side lengths$ AB = 19$, $BC = 180$, and $AC = 181$, and angle measure $\angle ABC = 90^o$. Let the midpoints of $AB$ and $BC$ be denoted by $M$ and $N$ respectively. The circle centered at $ M$ and passing through point $C$ intersects with the circle centered at the $N$ and passing through point $A$ at points $D$ and $E$. If $DE$ intersects $AC$ at point $P$, find min $(DP,EP)$.
2023 CUBRMC, 6
Find the sum of all positive divisors of $40081$.
2023 CUBRMC, 5
The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$. What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set?
2023 CUBRMC, 1
Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$-digit integer. Find all possible $1$-digit integers Ben can end with from this process.
2023 CUBRMC, 3
Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.
2023 CUBRMC, 9
Find the sum of all integers $n$ such that $1 < n < 30$ and $n$ divides
$$1+\sum^{n-1}_{k=1}k^{2k}.$$
2023 CUBRMC, 2
This season, there are $3n + 1$ teams in the MLS (Major League Soccer). As of now, each team has played exactly $n -1$ matches. Prove that there exist $4$ teams such that none of the $4$ teams have faced each other.
2023 CUBRMC, 3
Two buckets each have four balls; two red balls and two white balls in the first, and two red balls and two blue balls in the second. At first, a bucket is selected, then a ball in the bucket is selected, with both buckets and balls inside the selected bucket having equal probability of being chosen. Then, without replacement of the first ball, the process is repeated once more. Determine the probability that the first ball drawn being red if the second ball drawn was blue.
2023 CUBRMC, 4
Alice, Bob, Carol, and David decide that they will share meals and that one of them will cook each night. Because David enjoys cooking, he will cook on $4$ days of the week, while Alice, Bob, and Carol each pick a day of the week to cook on. If Alice, Bob, and Carol each choose the day they cook uniformly at random so as to avoid overlap, what is the probability that David does not cook on three consecutive days? For example, Monday, Tuesday and Wednesday are considered as three consecutive days, so are Saturday, Sunday and Monday.
2023 CUBRMC, 7
Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$.
2023 CUBRMC, 2
The concave decagon shown below is embedded in the Cartesian coordinate plane such that all of its vertices have integer coordinates. Two opposite edges have length $5$, whereas the remaining eight edges have length $\sqrt{10}$. Every pair of opposite edges is parallel. The sides of the decagon do not intersect each other, and the decagon has vertical and horizontal axes of symmetry. Find the area of the decagon.
[img]https://cdn.artofproblemsolving.com/attachments/1/5/daa4ab3d71af4b3274cd222f9a091eea3be705.png[/img]
2023 CUBRMC, 8
If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.
2023 CUBRMC, 4
Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.