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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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0 replies
jlacosta
May 1, 2025
0 replies
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[PMO Area Stage I. #4] logs + geo
ACalculationError   0
26 minutes ago
Let \( f(x) = \log_a x \) for some base \( a > 0 \), \( a \neq 1 \). The points \( (3, m) \), \( (x_1, y_1) \), and \( (x_2, y_2) \) lie on the graph of \( f \). If \(y_1 + y_2 = 2m\), find the value of \( x_1 x_2 \).
0 replies
ACalculationError
26 minutes ago
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a tst 2013 test
Math2030   1
N an hour ago by Math2030
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
1 reply
Math2030
Today at 5:26 AM
Math2030
an hour ago
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[Sipnayan JHS] Semifinals Round B, Average, #2
LilKirb   1
N 2 hours ago by LilKirb
How many trailing zeroes are there in the base $4$ representation of $2015!$ ?
1 reply
LilKirb
3 hours ago
LilKirb
2 hours ago
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2022 SMT Team Round - Stanford Math Tournament
parmenides51   5
N 3 hours ago by vanstraelen
p1. Square $ABCD$ has side length $2$. Let the midpoint of $BC$ be $E$. What is the area of the overlapping region between the circle centered at $E$ with radius $1$ and the circle centered at $D$ with radius $2$? (You may express your answer using inverse trigonometry functions of noncommon values.)


p2. Find the number of times $f(x) = 2$ occurs when $0 \le x \le 2022 \pi$ for the function $f(x) = 2^x(cos(x) + 1)$.


p3. Stanford is building a new dorm for students, and they are looking to offer $2$ room configurations:
$\bullet$ Configuration $A$: a one-room double, which is a square with side length of $x$,
$\bullet$ Configuration $B$: a two-room double, which is two connected rooms, each of them squares with a side length of $y$.
To make things fair for everyone, Stanford wants a one-room double (rooms of configuration $A$) to be exactly $1$ m$^2$ larger than the total area of a two-room double. Find the number of possible pairs of side lengths $(x, y)$, where $x \in N$, $y \in N$, such that $x - y < 2022$.


p4. The island nation of Ur is comprised of $6$ islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands $A$ and $B$ if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?


p5. Let $a, b,$ and $c$ be the roots of the polynomial $x^3 - 3x^2 - 4x + 5$. Compute $\frac{a^4 + b^4}{a + b}+\frac{b^4 + c^4}{b + c}+\frac{c^4 + a^4}{c + a}$.


p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.)


p7. Consider the sequence of integers an defined by $a_1 = 1$, $a_p = p$ for prime $p$ and $a_{mn} = ma_n + na_m$ for $m, n > 1$. Find the smallest $n$ such that $\frac{a_n^2}{2022}$ is a perfect power of $3$.


p8. Let $\vartriangle ABC$ be a triangle whose $A$-excircle, $B$-excircle, and $C$-excircle have radii $R_A$, $R_B$, and $R_C$, respectively. If $R_AR_BR_C = 384$ and the perimeter of $\vartriangle ABC$ is $32$, what is the area of $\vartriangle ABC$?


p9. Consider the set $S$ of functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 243\}$ satisfying:
(a) $f(1) = 1$
(b) $f(n^2) = n^2f(n)$,
(c) $n |f(n)$,
(d) $f(lcm(m, n))f(gcd(m, n)) = f(m)f(n)$.
If $|S|$ can be written as $p^{\ell_1}_1 \cdot p^{\ell_2}_2 \cdot ... \cdot p^{\ell_k}_k$ where $p_i$ are distinct primes, compute $p_1\ell_1+p_2\ell_2+. . .+p_k\ell_k$.


p10. You are given that $\log_{10}2 \approx 0.3010$ and that the first (leftmost) two digits of $2^{1000}$ are 10. Compute the number of integers $n$ with $1000 \le n \le 2000$ such that $2^n$ starts with either the digit $8$ or $9$ (in base $10$).


p11. Let $O$ be the circumcenter of $\vartriangle ABC$. Let $M$ be the midpoint of $BC$, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively, onto the opposite sides. $EF$ intersects $BC$ at $P$. The line passing through $O$ and perpendicular to $BC$ intersects the circumcircle of $\vartriangle ABC$ at $L$ (on the major arc $BC$) and $N$, and intersects $BC$ at $M$. Point $Q$ lies on the line $LA$ such that $OQ$ is perpendicular to $AP$. Given that $\angle BAC = 60^o$ and $\angle AMC = 60^o$, compute $OQ/AP$.


p12. Let $T$ be the isosceles triangle with side lengths $5, 5, 6$. Arpit and Katherine simultaneously choose points $A$ and $K$ within this triangle, and compute $d(A, K)$, the squared distance between the two points. Suppose that Arpit chooses a random point $A$ within $T$ . Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of $d(A, K)$. Compute this value.


p13. For a regular polygon $S$ with $n$ sides, let $f(S)$ denote the regular polygon with $2n$ sides such that the vertices of $S$ are the midpoints of every other side of $f(S)$. Let $f^{(k)}(S)$ denote the polygon that results after applying f a total of k times. The area of $\lim_{k \to \infty} f^{(k)}(P)$ where $P$ is a pentagon of side length $1$, can be expressed as $\frac{a+b\sqrt{c}}{d}\pi^m$ for some positive integers $a, b, c, d, m$ where $d$ is not divisible by the square of any prime and $d$ does not share any positive divisors with $a$ and $b$. Find $a + b + c + d + m$.


p14. Consider the function $f(m) = \sum_{n=0}^{\infty}\frac{(n - m)^2}{(2n)!}$ . This function can be expressed in the form $f(m) = \frac{a_m}{e} +\frac{b_m}{4}e$ for sequences of integers $\{a_m\}_{m\ge 1}$, $\{b_m\}_{m\ge 1}$. Determine $\lim_{n \to \infty}\frac{2022b_m}{a_m}$.


p15. In $\vartriangle ABC$, let $G$ be the centroid and let the circumcenters of $\vartriangle BCG$, $\vartriangle CAG$, and $\vartriangle ABG$ be $I, J$, and $K$, respectively. The line passing through $I$ and the midpoint of $BC$ intersects $KJ$ at $Y$. If the radius of circle $K$ is $5$, the radius of circle $J$ is $8$, and $AG = 6$, what is the length of $KY$ ?



PS. You should use hide for answers. Collected here.
5 replies
parmenides51
Jun 30, 2022
vanstraelen
3 hours ago
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ez problem....
Cobedangiu   4
N Apr 19, 2025 by iniffur
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
4 replies
Cobedangiu
Apr 18, 2025
iniffur
Apr 19, 2025
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ez problem....
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Cobedangiu
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Cobedangiu
#1 h  Apr 18, 2025, 11:07 AM
Y by
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
This post has been edited 1 time. Last edited by Cobedangiu, Apr 18, 2025, 11:08 AM
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Cobedangiu
70 posts
Cobedangiu
#2 h  Apr 19, 2025, 7:43 AM
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Cobedangiu wrote:
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$

no :<?
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Lankou
1405 posts
Lankou
#3 h  Apr 19, 2025, 4:06 PM
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What does the symbol $\cancel\vdots$ mean?
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giangtruong13
148 posts
giangtruong13
#4 h  Apr 19, 2025, 4:09 PM • 1 Y
Y by Lankou
Lankou wrote:
What does the symbol $\cancel\vdots$ mean?
It means $ 7 \cancel | xy $
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iniffur
538 posts
iniffur
#5 h  Apr 19, 2025, 11:22 PM
Y by
$$x^2+xy+y^2=7^n~~~~~~~~~~~~~~~~~~(1)$$
In its present wording the problem does not require to find out the triplets $~(x,~y,n)~$,

but rather the powers of seven which can be represented as $~~x^2+xy+y^2,~~$ with the proviso that

$~7\nmid xy~~$

It can be derived from the literature regarding $~~x^2+xy+y^2=N~~$ that if $~(x,y)=(a,b),~(c,d)~~$ are

two solutions to $~x^2+xy+y^2=7^n,~~~(x,y)=(ac+bc+bd,~ad-bc)~~$ is a further solution by virtue

of the multiplicative property of numbers under the form $~~x^2+xy+y^2=N~~$ (see (*) below)

Therefore, it is sufficient to depart from two solutions, one selected from the group consisting of:

$x^2+xy+y^2=7~~\Longrightarrow (a,b)=(1,2), (2,1), (-1,3), (-2,3), (-3,2), (-3,1), (-2, -1), (-1,-2), (1,-3), (2, -3), (3,-2),$

$(3, -1)$

and the second one selected from the group consisting of:

$ x^2+xy+y^2=49~~\Longrightarrow (c,d)= (3,5), (5,3), (-3,8), (-5,8), (-8,5), (-8,3), (-5,-3),(-3,-5), (-8,3), (8,-5)$

to derive further solutions corresponding to any power of seven by applying the formula:

$(x,y)=(ac+bc+bd,~ad-bc)$

Of course, care should be taken that $~~7\nmid ac+bc+bd,~ad-bc~~$

Example 1

$a=1, b=2, c=3, d=5\Longrightarrow (1*3+2*3+2*5=19,~~1*5-2*3=-1)\Longrightarrow 19^2-19+1=343=7^3$

Example 2

$a=-3, b=1, c=19, d=-1\Longrightarrow (-39,-16)\Longrightarrow 39^2+39*16+16^2=2401=7^4$

Example 3

$a=8, b=-5, c=19, d=-1\Longrightarrow (62,~87)\Longrightarrow 62^2+62*87+87^2=16807=7^5$

And so on.

It might be possible to generalize this approach to the power of other primes (at first sight 3 and 13 could

work).



(*)Click to reveal hidden text
This post has been edited 1 time. Last edited by iniffur, Apr 20, 2025, 1:12 PM
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