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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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A problem of collinearity.
Raul_S_Baz   2
N 3 minutes ago by Raul_S_Baz
Î am the author.
IMAGE
P.S: How can I verify that it is an original problem? Thanks!
2 replies
Raul_S_Baz
Yesterday at 4:19 PM
Raul_S_Baz
3 minutes ago
J
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Inequalities
sqing   0
28 minutes ago
Let $ a,b,c>0 $ . Prove that
$$\frac{a+5b}{b+c}+\frac{b+5c}{c+a}+\frac{c+5a}{a+b}\geq 9$$$$ \frac{2a+11b}{b+c}+\frac{2b+11c}{c+a}+\frac{2c+11a}{a+b}\geq \frac{39}{2}$$$$ \frac{25a+147b}{b+c}+\frac{25b+147c}{c+a}+\frac{25c+147a}{a+b} \geq258$$
0 replies
1 viewing
sqing
28 minutes ago
0 replies
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Plz help
Bet667   3
N Yesterday at 6:50 PM by K1mchi_
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)
3 replies
Bet667
Jan 28, 2024
K1mchi_
Yesterday at 6:50 PM
J
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2019 SMT Team Round - Stanford Math Tournament
parmenides51   17
N Yesterday at 6:40 PM by Rombo
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.
17 replies
parmenides51
Feb 6, 2022
Rombo
Yesterday at 6:40 PM
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No more topics!
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No bash for this inequality
giangtruong13   2
N Apr 9, 2025 by giangtruong13
Let $x,y,z$ be positive real number satisfy that: $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$.Find the minimum: $$ \sum_{cyc} \frac{(xy)^2}{z(x^2+y^2)} $$
2 replies
giangtruong13
Apr 8, 2025
giangtruong13
Apr 9, 2025
J
No bash for this inequality
G H J
Reveal topic tags
Inequality
inequalities
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giangtruong13
146 posts
giangtruong13
#1 h  Apr 8, 2025, 3:08 PM
Y by
Let $x,y,z$ be positive real number satisfy that: $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$.Find the minimum: $$ \sum_{cyc} \frac{(xy)^2}{z(x^2+y^2)} $$
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no_room_for_error
334 posts
no_room_for_error
#2 h  Apr 8, 2025, 5:41 PM
Y by
giangtruong13 wrote:
Let $x,y,z$ be positive real number satisfy that: $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$.Find the minimum: $$ \sum_{cyc} \frac{(xy)^2}{z(x^2+y^2)} $$

$$\frac{(xy)^2}{z(x^2+y^2)}=\frac{z}{z^2-1}\geq \frac{3\sqrt{3}}{2z^2}$$
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giangtruong13
146 posts
giangtruong13
#3 h  Apr 9, 2025, 11:30 PM
Y by
Nah this is bash
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