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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Problems with Progression and Series
SomeonecoolLovesMaths   13
N 3 minutes ago by SomeonecoolLovesMaths
These are a few questions I wasn't able to solve, any help will be appreciated!

$1.$ If $a_{n+1} = \frac{1}{1- a_n}$ for $n \geq 1$ and $a_3 = a_1$, then find the value of $(a_{2001})^{2001}$.

$2.$ If the $p$th term of an A.P. is $q$ and the $q$th term is $p$, then find its $r$th term.

$3.$ If $x$ is a positive real number different from $1$, then prove that the numbers $\frac{1}{1 + \sqrt{x}}, \frac{1}{1-x} , \frac{1}{1- \sqrt{x}}, \cdots$ are in A.P. Also find their common difference.

My Progress
13 replies
SomeonecoolLovesMaths
4 hours ago
SomeonecoolLovesMaths
3 minutes ago
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The six faces of a cube are painted in a manner that no two adjacent faces have
Vulch   0
3 hours ago
The six faces of a cube are painted in a manner that no two adjacent faces have the same colour.The three colours used in painting are red, blue and green.The cube is then cut into 36 smaller cubes in a manner that 32 cubes are of one size and the rest of a bigger size and each of the bigger cube has no red side.How many cubes only have one side coloured?
0 replies
Vulch
3 hours ago
0 replies
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2012 RMT Team Round - Stanford Math Tournament
parmenides51   7
N 3 hours ago by soryn
p1. How many functions $f : \{1, 2, 3, 4, 5\} \to \{1, 2, 3, 4, 5\}$ take on exactly $3$ distinct values?


p2. Let $i$ be one of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$. Suppose that for all positive integers $n$, the number $n^n$ never has remainder $i$ upon division by $12$. List all possible values of $i$.


p3. A card is an ordered 4-tuple $(a_1, a_2, a_3, a_4)$ where each $a_i$ is chosen from $\{0, 1, 2\}$. A line is an (unordered) set of three (distinct) cards $\{(a_1, a_2, a_3, a_4)$,$(b_1, b_2, b_3, b_4)$,$(c_1, c_2, c_3, c_4)\}$ such that for each $i$, the numbers $a_i, b_i, c_i$ are either all the same or all different. How many different lines are there?


p4. We say that the pair of positive integers $(x, y)$, where $x < y$, is a $k$-tangent pair if we have
$\arctan \frac{1}{k} = arctan\frac{1}{x}+ arctan\frac{1}{y}$ . Compute the second largest integer that appears in a $2012$-tangent pair.


p5. Regular hexagon $A_1A_2A_3A_4A_5A_6$ has side length $1$. For $i = 1, ..., 6$, choose $B_i$ to be a point on the segment $A_iA_{i+1}$ uniformly at random, assuming the convention that $A_{j+6} = A_j$ for all integers $j$. What is the expected value of the area of hexagon $B_1B_2B_3B_4B_5B_6$?


p6. Evaluate $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{nm(n + m + 1)}$.


p7. A plane in $3$-dimensional space passes through the point $(a_1, a_2, a_3)$, with $a_1$, $a_2$, and $a_3$ all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers $b_1$, $b_2$, $b_3$ such that $(b_1, 0, 0)$, $(0, b_2, 0)$, and $(0, 0, b_3)$ all lie on this plane). Find, in terms of $a_1$, $a_2$, $a_3$, the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.


p8. The left end of a rubber band e meters long is attached to a wall and a slightly sadistic child holds on to the right end. A point-sized ant is located at the left end of the rubber band at time $t = 0$, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of $1/(ln(t + e))$ centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer $+\infty$.


p9. We say that two lattice points are neighboring if the distance between them is $1$. We say that a point lies at distance d from a line segment if $d$ is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point $A$ is nearby a line segment if the distance between $A$ and the line segment is no greater than the distance between the line segment and any neighbor of $A$. Find the number of lattice points that are nearby the line segment connecting the origin and the point $(1984, 2012)$.


p10. A permutation of the first n positive integers is valid if, for all $i > 1$, $i$ comes after $\left\lfloor \frac{i}{2} \right\rfloor $ in the permutation. What is the probability that a random permutation of the first $14$ integers is valid?


p11. Given that $x, y, z > 0$ and $xyz = 1$, find the range of all possible values of
$\frac{x^3 + y^3 + z^3 - x^{-3} - y^{-3} - z^{-3}}{x + y + z - x^{-1} - y^{-1} - z^{-1}}$.


p12. A triangle has sides of length $\sqrt2$, $3 + \sqrt3$, and $2\sqrt2 + \sqrt6$. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle.


p13. How many positive integers $n$ are there such that for any natural numbers $a, b$, we have $n | (a^2b + 1)$ implies $n | (a^2 + b)$?


p14. Find constants $a$ and $c$ such that the following limit is finite and nonzero: $c = \lim_{n \to \infty} \frac{e\left( 1- \frac{1}{n}\right)^n - 1}{n^a}$.
Give your answer in the form $(a, c)$.


p15. Sean thinks packing is hard, so he decides to do math instead. He has a rectangular sheet that he wants to fold so that it fits in a given rectangular box. He is curious to know what the optimal size of a rectangular sheet is so that it’s expected to fit well in any given box. Let a and b be positive reals with $a \le b$, and let $m$ and $n$ be independently and uniformly distributed random variables in the interval $(0, a)$. For the ordered $4$-tuple $(a, b, m, n)$, let $f(a, b, m, n)$ denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension $m \times n$ after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation $f$.

PS. You had better use hide for answers.
7 replies
parmenides51
Jan 24, 2022
soryn
3 hours ago
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inequalities 070425
pennypc123456789   5
N Today at 6:21 AM by Sadigly
Let $a,b,c$ be positive real numbers . Prove that :
$$\dfrac{2ab}{a^2+b^2} + \dfrac{2bc}{b^2+c^2} + \dfrac{2ac}{a^2+c^2} \ge \dfrac{24abc}{(a+b)(b+c)(a+c)} $$
5 replies
pennypc123456789
Today at 4:24 AM
Sadigly
Today at 6:21 AM
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a+b+c=3 inequality
JK1603JK   0
Apr 2, 2025
Let $a,b,c\ge 0: a+b+c=3$ then prove
$$\color{black}{\sqrt{a+b+2c^{2}}+\sqrt{b+c+2a^{2}}+\sqrt{c+a+2b^{2}}\le 3\sqrt{\frac{a^2+b^2+c^2}{ab+bc+ca}+3}.}$$
0 replies
JK1603JK
Apr 2, 2025
0 replies
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a+b+c=3 inequality
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JK1603JK
#1 h  Apr 2, 2025, 11:43 AM
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Let $a,b,c\ge 0: a+b+c=3$ then prove
$$\color{black}{\sqrt{a+b+2c^{2}}+\sqrt{b+c+2a^{2}}+\sqrt{c+a+2b^{2}}\le 3\sqrt{\frac{a^2+b^2+c^2}{ab+bc+ca}+3}.}$$
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