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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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Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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
Random concyclicity in a square config
Maths_VC   3
N 22 minutes ago by Assassino9931
Source: Serbia JBMO TST 2025, Problem 1
Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$, and let $N$ be the intersection point of segments $AC$ and $DM$. The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$, $C$, $P$ and $Q$ lie on a single circle.
3 replies
Maths_VC
Tuesday at 7:38 PM
Assassino9931
22 minutes ago
Basic ideas in junior diophantine equations
Maths_VC   2
N 24 minutes ago by Assassino9931
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
2 replies
1 viewing
Maths_VC
Tuesday at 7:54 PM
Assassino9931
24 minutes ago
Prime number theory
giangtruong13   2
N 25 minutes ago by RagvaloD
Find all prime numbers $p,q$ such that: $p^2-pq-q^3=1$
2 replies
giangtruong13
an hour ago
RagvaloD
25 minutes ago
An algorithm for discovering prime numbers?
Lukaluce   2
N 27 minutes ago by Assassino9931
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
2 replies
Lukaluce
May 18, 2025
Assassino9931
27 minutes ago
Find the volume
son2007vn   0
May 26, 2025
In 3D space \( Oxyz \), a cone \( (N) \) has a vertex at \( A(1; 1; 1) \), and its base is a circle lying on the plane
$(P): x + 2y - z + 4 = 0$. The angle at the cone's vertex is \( 60^\circ \). Write the equation of a plane \( (Q) \) that contains the x-axis and divides the cone into two parts with a volume ratio of \( \dfrac{1}{7} \) (the part containing vertex \( A \) can have the smaller volume).
0 replies
son2007vn
May 26, 2025
0 replies
shadow of a cylinder, shadow of a cone
vanstraelen   4
N May 19, 2025 by mathafou

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
4 replies
vanstraelen
May 9, 2025
mathafou
May 19, 2025
Concurrent in a pyramid
vanstraelen   1
N May 17, 2025 by vanstraelen

Given a pyramid $(T,ABCD)$ where $ABCD$ is a parallelogram.
The intersection of the diagonals of the base is point $S$.
Point $A$ is connected to the midpoint of $[CT]$, point $B$ to the midpoint of $[DT]$,
point $C$ to the midpoint of $[AT]$ and point $D$ to the midpoint of $[BT]$.
a) Prove: the four lines are concurrent in a point $P$.
b) Calulate $\frac{TS}{TP}$.
1 reply
vanstraelen
May 10, 2025
vanstraelen
May 17, 2025
Trunk of cone
soruz   1
N May 14, 2025 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
May 14, 2025
Tetrahedron
4everwise   3
N May 10, 2025 by aidan0626
Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals

$\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6$
3 replies
4everwise
Jan 1, 2006
aidan0626
May 10, 2025
Triangle on a tetrahedron
vanstraelen   2
N May 9, 2025 by ReticulatedPython

Given a regular tetrahedron $(A,BCD)$ with edges $l$.
Construct at the apex $A$ three perpendiculars to the three lateral faces.
Take a point on each perpendicular at a distance $l$ from the apex such that these three points lie above the apex.
Calculate the lenghts of the sides of the triangle.
2 replies
vanstraelen
May 9, 2025
ReticulatedPython
May 9, 2025
Cube Sphere
vanstraelen   4
N May 9, 2025 by pieMax2713

Given the cube $\left(\begin{array}{ll} EFGH \\ ABCD \end{array}\right)$ with edge $6$ cm.
Find the volume of the sphere passing through $A,B,C,D$ and tangent to the plane $(EFGH)$.
4 replies
vanstraelen
May 9, 2025
pieMax2713
May 9, 2025
parallelogram in a tetrahedron
vanstraelen   1
N May 9, 2025 by vanstraelen
Given a tetrahedron $ABCD$ and a plane $\mu$, parallel with the edges $AC$ and $BD$.
$AB \cap \mu=P$.
a) Prove: the intersection of the tetrahedron with the plane is a parallelogram.
b) If $\left|AC\right|=14,\left|BD\right|=7$ and $\frac{\left|PA\right|}{\left|PB\right|}=\frac{3}{4}$,
calculates the lenghts of the sides of this parallelogram.
1 reply
vanstraelen
May 5, 2025
vanstraelen
May 9, 2025
Regular tetrahedron
vanstraelen   7
N May 6, 2025 by ReticulatedPython
Given the points $O(0,0,0),A(1,0,0),B(\frac{1}{2},\frac{\sqrt{3}}{2},0)$
a) Determine the point $C$, above the xy-plane, such that the pyramid $OABC$ is a regular tetrahedron.
b) Calculate the volume.
c) Calculate the radius of the inscribed sphere and the radius of the circumscribed sphere.
7 replies
vanstraelen
May 4, 2025
ReticulatedPython
May 6, 2025
volume 9f a pentagonal base pyramid circumscribed around a right circular cone
FOL   1
N May 6, 2025 by Mathzeus1024
A pentagonal base pyramid is circumscribed around a right circular cone, whose height is equal to the radius of the base. The total surface area of the pyramid is d times greater than that of the cone. Find the volume of the pyramid if the lateral surface area of the cone is equal to $\pi\sqrt{2}$.
1 reply
FOL
Jul 22, 2023
Mathzeus1024
May 6, 2025
Polynomials
P162008   1
N Apr 25, 2025 by thehound
Define a family of polynomials by $P_{0}(x) = x - 2$ and $P_{k}(x) = \left(P_{k - 1} (x)\right)^2 - 2$ if $k \geq 1$ then find the coefficient of $x^2$ in $P_{k}(x)$ in terms of $k.$
1 reply
P162008
Apr 25, 2025
thehound
Apr 25, 2025
Polynomials
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P162008
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Define a family of polynomials by $P_{0}(x) = x - 2$ and $P_{k}(x) = \left(P_{k - 1} (x)\right)^2 - 2$ if $k \geq 1$ then find the coefficient of $x^2$ in $P_{k}(x)$ in terms of $k.$
This post has been edited 2 times. Last edited by P162008, Apr 25, 2025, 2:06 AM
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thehound
12 posts
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The answer is $\frac{4^{k-1}(4^k - 1)}{3}$. Call $L_k, C_k, Q_k$ the linear, constant, and quadratic coefficients respectively of $P_k(x)$. Clearly, $Q_k = 2Q_{k - 1}C_{k - 1} + L_{k - 1}^2$ for all $k \geq 1$ by polynomial multiplication. It is easy to show that $C_k = 2$ for all $k \geq 1$ using induction since $C_1 = 2$. This combined with the fact that $Q_0 = 0$ and $L_0 = 1$ gives us $Q_k = 4Q_{k - 1} + L_{k - 1}^2$ for all $k \geq 1$. Additionally, $L_k = 2L_{k - 1}C_{k - 1} = 4L_{k - 1}$ for all $k \geq 2$ and since $L_1 = -4$, $L_k = -4^k$ for all $k \geq 1$. This combined with $L_0 = 1$ and $Q_0 = 0$ gives $Q_k = 4Q_{k - 1} + L_{k - 1}^2 = 4Q_{k - 1} + 4^{2k - 2}$. We will show next with induction that $Q_k = \frac{4^{k-1}(4^k - 1)}{3}$ for all $k \geq 0$.

Base Case: Since k = 0, $\frac{4^{0 - 1}(4^0 - 1)}{3} = 0$ which is true since $P_0$ is linear.
Inductive Step: I.H.: Assume the formula holds true for k. Since $k + 1 \geq 1$, $Q_{k  + 1} = 4Q_{k} + 4^{2k}$. By I.H., $Q_{k + 1} = 4 \cdot \frac{4^{k - 1}(4^k - 1)}{3} + 4^{2k} = \frac{4^k(4^k - 1) + 3 \cdot 4^{2k}}{3} = \frac{4^k((4 - 1) \cdot 4^k + 4^k - 1)}{3} = \frac{4^k(4^{k + 1} - 4^k + 4^k - 1)}{3} = \frac{4^k(4^{k + 1} - 1)}{3}$ as desired. By principle of mathematical induction, the formula holds true for all $k \geq 0$.
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