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Contests & Programs AMC and other contests, summer programs, etc.

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Contests & Programs AMC and other contests, summer programs, etc.

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Alcumus vs books

UnbeatableJJ 5

N 3 hours ago by Andyluo

If I am aiming for AIME, then JMO afterwards, is Alcumus adequate, or I still need to do the problems on AoPS books?

I got AMC 23 this year, and never took amc 10 before. If I master the alcumus of intermediate algebra (making all of the bars blue). How likely I can qualify for AIME 2026?

5 replies

UnbeatableJJ

Apr 23, 2025

Andyluo

3 hours ago

Find function

trito11 3

N Yesterday at 8:37 PM by jasperE3

Find $f:\mathbb{R^+} \to \mathbb{R^+}$ such that
i) f(x)>f(y) $\forall$ x>y>0
ii) f(2x) $\ge$ 2f(x) $\forall$ x>0
iii) $f(f(x)f(y)+x)=f(xf(y))+f(x)$ $\forall$ x,y>0

3 replies

trito11

Nov 11, 2019

jasperE3

Yesterday at 8:37 PM

2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)

parmenides51 1

N Yesterday at 8:09 PM by NamelyOrange

Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^{2024}$ and $128$ divides
$n^{\sigma (n)} - \sigma(n^n)$ where $\sigma (n)$ denotes the number of positive integers that divide $n$ , including $1$ and $n$ . Find the remainder when $N$ is divided by $1000$ .

1 reply

parmenides51

Jan 29, 2025

NamelyOrange

Yesterday at 8:09 PM

Inequalities

sqing 2

N Yesterday at 7:59 PM by maromex

Let $a,b,c>0$ . Prove that
$\frac{a+5b}{b+c}+\frac{b+5c}{c+a}+\frac{c+5a}{a+b}\geq 9$ $\frac{2a+11b}{b+c}+\frac{2b+11c}{c+a}+\frac{2c+11a}{a+b}\geq \frac{39}{2}$ $\frac{25a+147b}{b+c}+\frac{25b+147c}{c+a}+\frac{25c+147a}{a+b} \geq258$

2 replies

sqing

Yesterday at 3:46 AM

maromex

Yesterday at 7:59 PM

Assam Mathematics Olympiad 2022 Category III Q14

SomeonecoolLovesMaths 2

N Yesterday at 7:21 PM by rachelcassano

The following sum of three four digits numbers is divisible by $75$ , $7a71 + 73b7 + c232$ , where $a, b, c$ are decimal digits. Find the necessary conditions in $a, b, c$ .

2 replies

SomeonecoolLovesMaths

Sep 12, 2024

rachelcassano

Yesterday at 7:21 PM

2019 SMT Team Round - Stanford Math Tournament

parmenides51 19

N Yesterday at 5:21 PM 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

Yesterday at 5:21 PM

Complex Number Geometry

gauss202 1

N Yesterday at 3:08 PM by ANewName

Describe the locus of complex numbers, $z$ , such that $\arg \left(\dfrac{z+i}{z-1} \right) = \dfrac{\pi}{4}$ .

1 reply

gauss202

Yesterday at 12:21 PM

ANewName

Yesterday at 3:08 PM

Inequalities

sqing 5

N Yesterday at 2:49 PM by sqing

Let $a,b,c >2$ and $ab+bc+ca \leq 75.$ Show that
$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$ Let $a,b,c >2$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$

5 replies

sqing

Tuesday at 11:31 AM

sqing

Yesterday at 2:49 PM

Trig Identity

gauss202 1

N Yesterday at 12:33 PM by Lankou

Simplify $\dfrac{1-\cos \theta + \sin \theta}{\sqrt{1 - \cos \theta + \sin \theta - \sin \theta \cos \theta}}$

1 reply

gauss202

Yesterday at 12:12 PM

Lankou

Yesterday at 12:33 PM

Trunk of cone

soruz 1

N Yesterday at 9:59 AM by Mathzeus1024

One hemisphere is putting a truncated cone, with the base circles hemisphere. How height should have truncated cone as its lateral area to be minimal side?

1 reply

soruz

May 6, 2015

Mathzeus1024

Yesterday at 9:59 AM

Inequalities

sqing 7

N Yesterday at 8:29 AM by sqing

Let $a,b,c >1$ and $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=1.$ Show that $ab+bc+ca \geq 48$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{3}{4}$ Let $a,b,c >1$ and $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=2.$ Show that $ab+bc+ca \geq \frac{75}{4}$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{6}{5}$ Let $a,b,c >1$ and $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=3.$ Show that $ab+bc+ca \geq 12$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq \frac{3}{2}$

7 replies

sqing

Tuesday at 9:04 AM

sqing

Yesterday at 8:29 AM

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