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Polynomial Minimization

ReticulatedPython 4

N May 21, 2025 by jasperE3

Consider the polynomial $p(x)=x^{n+1}-x^{n}$ , where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$

4 replies

ReticulatedPython

May 6, 2025

jasperE3

May 21, 2025

2019 SMT Team Round - Stanford Math Tournament

parmenides51 19

N May 14, 2025 by SomeonecoolLovesMaths

p1. Given $x + y = 7$ , find the value of x that minimizes $4x^2 + 12xy + 9y^2$ .

p2. There are real numbers $b$ and $c$ such that the only $x$ -intercept of $8y = x^2 + bx + c$ equals its $y$ -intercept. Compute $b + c$ .

p3. Consider the set of $5$ digit numbers $ABCDE$ (with $A \ne 0$ ) such that $A+B = C$ , $B+C = D$ , and $C + D = E$ . What’s the size of this set?

p4. Let $D$ be the midpoint of $BC$ in $\vartriangle ABC$ . A line perpendicular to D intersects $AB$ at $E$ . If the area of $\vartriangle ABC$ is four times that of the area of $\vartriangle BDE$ , what is $\angle ACB$ in degrees?

p5. Define the sequence $c_0, c_1, ...$ with $c_0 = 2$ and $c_k = 8c_{k-1} + 5$ for $k > 0$ . Find $\lim_{k \to \infty} \frac{c_k}{8^k}$ .

p6. Find the maximum possible value of $|\sqrt{n^2 + 4n + 5} - \sqrt{n^2 + 2n + 5}|$ .

p7. Let $f(x) = \sin^8 (x) + \cos^8(x) + \frac38 \sin^4 (2x)$ . Let $f^{(n)}$ (x) be the $n$ th derivative of $f$ . What is the largest integer $a$ such that $2^a$ divides $f^{(2020)}(15^o)$ ?

p8. Let $R^n$ be the set of vectors $(x_1, x_2, ..., x_n)$ where $x_1, x_2,..., x_n$ are all real numbers. Let $||(x_1, . . . , x_n)||$ denote $\sqrt{x^2_1 +... + x^2_n}$ . Let $S$ be the set in $R^9$ given by $S = \{(x, y, z) : x, y, z \in R^3 , 1 = ||x|| = ||y - x|| = ||z -y||\}.$ If a point $(x, y, z)$ is uniformly at random from $S$ , what is $E[||z||^2]$ ?

p9. Let $f(x)$ be the unique integer between $0$ and $x - 1$ , inclusive, that is equivalent modulo $x$ to $\left( \sum^2_{i=0} {{x-1} \choose i} ((x - 1 - i)! + i!) \right)$ . Let $S$ be the set of primes between $3$ and $30$ , inclusive. Find $\sum_{x\in S}^{f(x)}$ .

p10. In the Cartesian plane, consider a box with vertices $(0, 0)$ , $\left( \frac{22}{7}, 0\right)$ , $(0, 24)$ , $\left( \frac{22}{7}, 4\right)$ . We pick an integer $a$ between $1$ and $24$ , inclusive, uniformly at random. We shoot a puck from $(0, 0)$ in the direction of $\left( \frac{22}{7}, a\right)$ and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at $(0, 0)$ and when it ends at some vertex of the box?

p11. Sarah is buying school supplies and she has $\$2019$ . She can only buy full packs of each of the following items. A pack of pens is $\$4$ , a pack of pencils is $\$3$ , and any type of notebook or stapler is $\$1$ . Sarah buys at least $1$ pack of pencils. She will either buy $1$ stapler or no stapler. She will buy at most $3$ college-ruled notebooks and at most $2$ graph paper notebooks. How many ways can she buy school supplies?

p12. Let $O$ be the center of the circumcircle of right triangle $ABC$ with $\angle ACB = 90^o$ . Let $M$ be the midpoint of minor arc $AC$ and let $N$ be a point on line $BC$ such that $MN \perp BC$ . Let $P$ be the intersection of line $AN$ and the Circle $O$ and let $Q$ be the intersection of line $BP$ and $MN$ . If $QN = 2$ and $BN = 8$ , compute the radius of the Circle $O$ .

p13. Reduce the following expression to a simplified rational $\frac{1}{1 - \cos \frac{\pi}{9}}+\frac{1}{1 - \cos \frac{5 \pi}{9}}+\frac{1}{1 - \cos \frac{7 \pi}{9}}$

p14. Compute the following integral $\int_0^{\infty} \log (1 + e^{-t})dt$ .

p15. Define $f(n)$ to be the maximum possible least-common-multiple of any sequence of positive integers which sum to $n$ . Find the sum of all possible odd $f(n)$

PS. You should use hide for answers. Collected here.

19 replies

parmenides51

Feb 6, 2022

SomeonecoolLovesMaths

May 14, 2025

Calculus BC help

needcalculusasap45 7

N Apr 20, 2025 by ehz2701

So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

7 replies

needcalculusasap45

Apr 19, 2025

ehz2701

Apr 20, 2025

Help with Competitive Geometry?

REACHAW 3

N Apr 15, 2025 by REACHAW

Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).

It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).

Do you have any advice, resources, or techniques I should try?

3 replies

REACHAW

Apr 14, 2025

REACHAW

Apr 15, 2025

geometry parabola problem

smalkaram_3549 10

N Apr 13, 2025 by ReticulatedPython

How would you solve this without using calculus?

10 replies

smalkaram_3549

Apr 11, 2025

ReticulatedPython

Apr 13, 2025

2012 RMT Team Round - Stanford Math Tournament

parmenides51 13

N Apr 9, 2025 by fruitmonster97

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.

13 replies

parmenides51

Jan 24, 2022

fruitmonster97

Apr 9, 2025

Solve the equetion

yt12 5

N Apr 3, 2025 by KevinKV01

Solve the equetion: $\sin 2x+\tan x=2$

5 replies

yt12

Mar 31, 2025

KevinKV01

Apr 3, 2025

Find the value

yt12 5

N Mar 20, 2025 by vanstraelen

Find the value of $\tan^6 ( \frac{\pi}{18}) + \tan^6(\frac{5 \pi}{18}) + \tan^6(\frac{7 \pi}{18})$

5 replies

yt12

Mar 16, 2025

vanstraelen

Mar 20, 2025

Finding Max/Min value with derivatives

POT4TO_71 4

N Mar 10, 2025 by Lankou

Knowing that the minimum/maximum value is the only point in a quadratic graph with gradient 0, we can use the latter to figure out the min/max value accordingly using basic derivatives; but do we prefer to do it this way or stick to completing the square?

Post opinions below :D

4 replies

POT4TO_71

Mar 9, 2025

Lankou

Mar 10, 2025

IOQM P10 2024

SomeonecoolLovesMaths 5

N Mar 7, 2025 by L13832

Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a,b,$ and $c$ which satisfy $a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$

5 replies

SomeonecoolLovesMaths

Sep 8, 2024

L13832

Mar 7, 2025

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