High School Math function

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Last Poster

Sequence

Titibuuu 1

N 40 minutes ago by Titibuuu

Let $a_1 = a$ , and for all $n \geq 1$ , define the sequence $\{a_n\}$ by the recurrence
$a_{n+1} = a_n^2 + 1$ Prove that there is no natural number $n$ such that
$\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)$ is a perfect square.

1 reply

Titibuuu

6 hours ago

Titibuuu

40 minutes ago

J

H

Show that three lines concur

benjaminchew13 2

N an hour ago by benjaminchew13

Source: Revenge JOM 2025 P2

t $A B C$ be a triangle. $M$ is the midpoint of segment $B C$ , and points $E$ , $F$ are selected on sides $A B$ , $A C$ respectively such that $E$ , $F$ , $M$ are collinear. The circumcircles $(A B C)$ and $(A E F)$ intersect at a point $P != A$ . The circumcircle $(A P M)$ intersects line $B C$ again at a point $D != M$ . Show that the lines $A D$ , $E F$ and the tangent to $(A E F)$ at point $P$ concur.

2 replies

benjaminchew13

an hour ago

benjaminchew13

an hour ago

J

H

slightly easy NT fe

benjaminchew13 2

N an hour ago by benjaminchew13

Source: Revenge JOM 2025 P1

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(a) + f(b) + f(c) | a^2 + af(b) + cf(a)$ for all $a, b, c\in\mathbb{N}$

2 replies

benjaminchew13

an hour ago

benjaminchew13

an hour ago

J

H

Cheesy's math casino

benjaminchew13 1

N an hour ago by benjaminchew13

Source: Revenge JOM 2025 P4

There are $p$ people playing a game at Cheesy's math casino, where $p$ is an odd prime number. Let $n$ be a positive integer. A subset of length $s$ from the set of integers from $1$ to $n$ inclusive is randomly chosen, with an equal probability ( $s <= n$ and is fixed). The winner of Cheesy's game is person $i$ , if the sum of the chosen numbers are congruent to $i mod p$ for $0 <= i <= p - 1$ .

For each $n$ , find all values of $s$ such that no one will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).

1 reply

benjaminchew13

an hour ago

benjaminchew13

an hour ago

J

H

2013 Japan MO Finals

parkjungmin 0

an hour ago

help me

we cad do it

0 replies

parkjungmin

an hour ago

0 replies

J

H

inequality

benjaminchew13 1

N an hour ago by benjaminchew13

Source: Revenge JOM 2025 P3

Let $n \ge 2$ be a positive integer and let $a_1$ , $a_2$ , ..., $a_n$ be a sequence of non-negative real numbers. Find the maximum value of $3(a_1 + a_2 + \cdots + a_n) - (a_1^2 + a_2^2 + \cdots + a_n^2) - (a_1a_2\cdots a_n)$ in terms of $n$ .

1 reply

benjaminchew13

an hour ago

benjaminchew13

an hour ago

J

H

IMO ShortList 1999, algebra problem 2

orl 11

N an hour ago by ezpotd

Source: IMO ShortList 1999, algebra problem 2

The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ( $n \geq 2$ ). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the characteristic of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?

11 replies

orl

Nov 14, 2004

ezpotd

an hour ago

J

H

Coolabra

Titibuuu 2

N an hour ago by no_room_for_error

Let $a, b, c$ be distinct real numbers such that
$a + b + c + \frac{1}{abc} = \frac{19}{2}$ Find the maximum possible value of $a$ .

2 replies

Titibuuu

6 hours ago

no_room_for_error

an hour ago

J

H

Hard centroid geo

lucas3617 0

an hour ago

Source: Revenge JOM 2025 P5

A triangle $A B C$ has centroid $G$ . A line parallel to $B C$ passing through $G$ intersects the circumcircle of $A B C$ at $D$ . Let lines $A D$ and $B C$ intersect at $E$ . Suppose a point $P$ is chosen on $B C$ such that the tangent of the circumcircle of $D E P$ at $D$ , the tangent of the circumcircle of $A B C$ at $A$ and $B C$ concur. Prove that $G P = P D$ .