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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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Original Question #2
Siopao_Enjoyer   1
N 3 hours ago by Siopao_Enjoyer
Let x, y, and z be positive real numbers such that:

√x + √y + √z = 17/3
1/√x + 1/√y + 1/√z = 21/4

If xyz = 16/9, one of the variables will be rational while the other two will be irrational. What is the value of that rational number?
1 reply
Siopao_Enjoyer
3 hours ago
Siopao_Enjoyer
3 hours ago
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[PMO17 Qualifying III.5] Roots a+2/a-2
LilKirb   1
N 3 hours ago by pooh123
Let $\alpha$, $\beta$, and $\gamma$ be the roots of $x^3 - 4x - 8 = 0.$ Find the numerical value of the expression:
\[\frac{\alpha + 2}{\alpha - 2} + \frac{\beta + 2}{\beta - 2} + \frac{\gamma + 2}{\gamma - 2}\]
1 reply
LilKirb
5 hours ago
pooh123
3 hours ago
J
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2017 Mathirang Mathibay - Orals, Tier 2 Easy
elpianista227   2
N 3 hours ago by anduran
Let $M, S, A$ be the roots of the polynomial $f(x) = 127x^3 + 1729x + 8128$. Find $(M + S)^3 + (S+ A)^3 + (A + M)^3$
2 replies
elpianista227
5 hours ago
anduran
3 hours ago
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24th PMO, Qualifying Stage #7
elpianista227   2
N 4 hours ago by tapilyoca
Suppose $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 2$. Let $f$ be the unique monic polynomial whose roots are $a^2, b^2, c^2$. Find $f(1)$.
2 replies
elpianista227
6 hours ago
tapilyoca
4 hours ago
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Quadruple Binomial Coefficient Sum
P162008   4
N Yesterday at 8:40 PM by vmene
Source: Self made by my Elder brother
$\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} \sum_{s=0}^{p+q - 1} \frac{((-1)^{p+r+s+1})(2^{p+q-1}) \binom{p + q - s - 1}{p + q - 2s - 1}}{4^s(2p^2q + 2pqr + pq + qr)(2p + 2q + 2r + 3)}.$
4 replies
P162008
Thursday at 8:04 PM
vmene
Yesterday at 8:40 PM
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IMC 1994 D1 P5
j___d   5
N Yesterday at 5:39 PM by krigger
a) Let $f\in C[0,b]$, $g\in C(\mathbb R)$ and let $g$ be periodic with period $b$. Prove that $\int_0^b f(x) g(nx)\,\mathrm dx$ has a limit as $n\to\infty$ and
$$\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx$$
b) Find
$$\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx$$
5 replies
j___d
Mar 6, 2017
krigger
Yesterday at 5:39 PM
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2023 Putnam A2
giginori   21
N Yesterday at 3:32 PM by pie854
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2 n$; that is to say, $p(x)=$ $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$ for some real coefficients $a_0, \ldots, a_{2 n-1}$. Suppose that $p(1 / k)=k^2$ for all integers $k$ such that $1 \leq|k| \leq n$. Find all other real numbers $x$ for which $p(1 / x)=x^2$.
21 replies
giginori
Dec 3, 2023
pie854
Yesterday at 3:32 PM
J
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Putnam 2019 A1
awesomemathlete   33
N Yesterday at 3:25 PM by cursed_tangent1434
Source: 2019 William Lowell Putnam Competition
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.
33 replies
awesomemathlete
Dec 10, 2019
cursed_tangent1434
Yesterday at 3:25 PM
J
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IMC 1994 D1 P2
j___d   5
N Yesterday at 3:11 PM by krigger
Let $f\in C^1(a,b)$, $\lim_{x\to a^+}f(x)=\infty$, $\lim_{x\to b^-}f(x)=-\infty$ and $f'(x)+f^2(x)\geq -1$ for $x\in (a,b)$. Prove that $b-a\geq\pi$ and give an example where $b-a=\pi$.
5 replies
j___d
Mar 6, 2017
krigger
Yesterday at 3:11 PM
J
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A Construction in Multivariable Analysis
MrOrange   0
Yesterday at 2:11 PM
Source: Garling's A COURSE IN MATHEMATICAL ANALYSIS
Construct a continuous real valued function \( f \) on \( \mathbb{R}^2 \) for which
\[ \lim_{R \to \infty} \int_{\|x\|_2 \leq R} f(x) \, dx = 0 \]and for which
\[ \lim_{R \to \infty} \int_{\|x\|_\infty \leq R} f(x) \, dx \text{ does not exist.} \]
0 replies
MrOrange
Yesterday at 2:11 PM
0 replies
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Possible values of determinant of 0-1 matrices
mathematics2004   4
N Yesterday at 1:56 PM by loup blanc
Source: 2021 Simon Marais, A3
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
Here $\det A$ denotes the determinant of the matrix $A$.
4 replies
mathematics2004
Nov 2, 2021
loup blanc
Yesterday at 1:56 PM
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ISI UGB 2025
Entrepreneur   1
N Yesterday at 1:49 PM by Knight2E4
Source: ISI UGB 2025
1.)
Suppose $f:\mathbb R\to\mathbb R$ is differentiable and $|f'(x)|<\frac 12\;\forall\;x\in\mathbb R.$ Show that for some $x_0\in\mathbb R,f(x_0)=x_0.$

3.)
Suppose $f:[0,1]\to\mathbb R$ is differentiable with $f(0)=0.$ If $|f'(x)|\le f(x)\;\forall\;x\in[0,1],$ then show that $f(x)=0\;\forall\;x.$

4.)
Let $S^1=\{z\in\mathbb C:|z|=1\}$ be the unit circle in the complex plane. Let $f:S^1\to S^1$ be the map given by $f(z)=z^2.$ We define $f^{(1)}:=f$ and $f^{(k+1)}=f\circ f^{(k)}$ for $k\ge 1.$ The smallest positive integer $n$ such that $f^n(z)=z$ is called period of $z.$ Determine the total number of points $S^1$ of period $2025.$

6.)
Let $\mathbb N$ denote the set of natural numbers, and let $(a_i,b_i), 1\le i\le 9,$ be nine distinct tuples in $\mathbb N\times\mathbb N.$ Show that there are $3$ distinct elements in the set $\{2^{a_i}3^{b_i}:1\le i\le 9\}$ whose product is a perfect cube.

8.)
Let $n\ge 2$ and let $a_1\le a_2\le\cdots\le a_n$ be positive integers such that $$\sum_{i=1}^n a_i=\prod_{i=1}^n a_i.$$Prove that $$\sum_{i=1}^n a_i\le 2n$$and determine when equality holds.
1 reply
Entrepreneur
May 27, 2025
Knight2E4
Yesterday at 1:49 PM
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Recurrence trouble
SomeonecoolLovesMaths   3
N Yesterday at 1:44 PM by Knight2E4
Let $0 < x_0 < y_0$ be real numbers. Define $x_{n+1} = \frac{x_n + y_n}{2}$ and $y_{n+1} = \sqrt{x_{n+1}y_n}$.
Prove that $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$ and hence find the limit.
3 replies
SomeonecoolLovesMaths
May 28, 2025
Knight2E4
Yesterday at 1:44 PM
J
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Trigo or Complex no.?
hzbrl   5
N Yesterday at 9:20 AM by GreenKeeper
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
5 replies
hzbrl
May 27, 2025
GreenKeeper
Yesterday at 9:20 AM
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geometry
carvaan   1
N Apr 20, 2025 by Lankou
The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.
1 reply
carvaan
Apr 20, 2025
Lankou
Apr 20, 2025
J
geometry
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carvaan
3 posts
carvaan
#1 h  Apr 20, 2025, 5:46 PM
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The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.
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Lankou
1406 posts
Lankou
#2 h  Apr 20, 2025, 6:38 PM
Y by
$\alpha-\beta=24$
if $\alpha$ and $\beta$ are double digits $(\alpha, \beta) \in \{(34, 10), (35, 11)....(99, 75)\}$
$\gamma=180-\alpha-\beta$
Since $\gamma$ is double digits, then the possibilities for $\alpha$ and $\beta$ must be reduced to $\{(53, 29), (54, 30), ....(97, 73)\}$
There are 45 possible values for the third angle
This post has been edited 1 time. Last edited by Lankou, Apr 20, 2025, 6:39 PM
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