AoPS Academy
+9 w
. Let 
be a point on side 
, and 
be a point on the extension of 
such that 
. Let 
be the circumcenter of 
, and 
be a point on the side extension of 
satisfying 
. Line BP intersects AC at point Q. Prove that 
be a prime and suppose 
is prime. Prove that 
is prime
This is a special case of Pocklington's theorem. A proof of this special case is required.
be the circumcenter and 
the orthocenter of an acute triangle 
. Show that there exist points 
, 
, and on sides , 
, and 
respectively such that ![\[ OD + DH = OE + EH = OF + FH\]](http://latex.artofproblemsolving.com/a/8/7/a874af4cb802b9b341b055140a08e096aaf2112e.png)
and the lines 
, 
, and 
are concurrent.
with 
. Denote the incircle of 
as 
and let meet 
at 
respectively. Draw circle 
such that passes through 
and tangents to at . Denote the intersections of line and as 
and 
, the length of 
can be written in the simplest form of 
, compute 
.
, and 
. The spheres with centers , and touch the plane at 
, and 
, respectively. Prove that 
of the pizza is covered by both toppings.
is a positive integer. For an -topping pizza, determine the probability, in terms of , that some region of the pizza with non-zero area is covered by all toppings.
liter and is initially filled with tea of 
% concentration. I want to serve tea to my guests, but they insist that the tea must have at least a 
% tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?
is the midpoint of and is the midpoint of 
. Suppose 
is similar to 
. Find the length of .
be a finite set of real numbers, 
be a real number, and 
be 2025 non-zero real numbers. Define![\[A =
\left\{
(x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d
\right\},\]](http://latex.artofproblemsolving.com/2/e/1/2e18b6d25bfaa3e3d52f7efe5e4cba7a9cb87db9.png)
![\[B =
\left\{
(x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0
\right\},\]](http://latex.artofproblemsolving.com/4/1/f/41f6b126cc3cf24757b37147906c767d49a0fdd9.png)
![\[C =
\left\{
(x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0
\right\}.\]](http://latex.artofproblemsolving.com/d/e/6/de6586b88004e3f5aa7774525b00ee975c072ff9.png)
Show that 
.
where 
. In a similar vein,
By Cauchy-Schwarz, we have
Now, the key is to take 2025 independent copies of this integral, each aptly rescaled:
where we used Holder's inequality and triangle inequality respectively.
denote 
, which is the constant term of 
. Then, if 
, then
and 
.
Equality holds when 
or 
or 
![\[
\begin{aligned}
&\text{Each question is worth 10 marks. If you just provide the answer, you get 5 marks. If you provide sufficient workings, you receive up to 5 marks.} \\[10pt]
&\textbf{Q1.} \text{ Alice and Bob are playing a game where Alice starts first. There is a common positive integer } x \text{ given, and on their turn,} \\
&\text{each player subtracts an integer } n \text{ where } 1 \leq n \leq 9, \text{ such that the common number becomes } (x-n). \text{ Given a target } y, \\
&\text{the player wins when their turn ends with } (x-n) = y. \\[10pt]
&\text{E.g. } x = 25, y = 1: \\
&\text{On Alice's turn, she chooses to subtract 9, so the common number is now 14.} \\
&\text{On Bob's turn, he chooses to subtract 3, so the common number is now 11.} \\
&\text{On Alice's turn, she chooses to subtract 2, so the common number is now 9.} \\
&\text{On Bob's turn, he chooses to subtract 8, so the common number is now 1. Bob wins.} \\[10pt]
&(i) \text{ Assuming } (x, y) \text{ is a pair of integers such that Alice will have a strategy to guarantee a win, find that strategy.} \\
&(ii) \text{ Find all } (x, y) \text{ where Bob will have a strategy to guarantee a win.} \\
&\text{[Modified Intermediate Mathematical Olympiad Maclaurin paper Q2]} \\[20pt]
&\textbf{Q2.} \text{ Given a fair } n\text{-sided die, where the sides are } 1, 2, 3, \ldots, n-1, n, \text{ find the probability of rolling } n \\
&\text{at least once in } m \text{ rolls.} \\
&\text{[Original question]} \\[20pt]
&\textbf{Q3.} \text{ Determine the smallest natural number } n \text{ such that } n^n \text{ is not a divisor of } 2025!. \\
&\text{[Modified Flanders Math Olympiad 2016 Q2]} \\[20pt]
&\textbf{Q4.} (n+1)^{n-1} = (n-1)^{n+1}. \text{ Find all real } n. \\
&\text{[Original Question]} \\[20pt]
&\textbf{Q5.} a, b, c, d, x \text{ are integers. } 0 \leq a, b, c, d \leq 9. \text{ Find the number of possible } (a, b, c, d) \text{ such that} \\
&7^a + 7^b + 7^c + 7^d = 100x. \\
&\text{Note: } (2, 0, 2, 4) \text{ and } (2, 0, 4, 2) \text{ are 2 separate solutions.} \\
&\text{[Intermediate Mathematical Olympiad Maclaurin paper Q3]} \\[20pt]
&\textbf{Q6.} \text{A sequence is defined as } a_1 = 2025, \text{ and for all } n \geq 2: \\
&a_n = \frac{a_{n-1} + 1}{n}. \\
&\text{Determine the smallest } k \text{ such that } a_k < \frac{1}{2025}. \\
&\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q1]} \\[20pt]
&\textbf{Q7.} \text{There are } n \geq 3 \text{ students in a classroom. Every day, the teacher splits the students into exactly 2 non-empty groups,} \\
&\text{and each pair of students from the same group will shake hands once. Suppose after } k \text{ days, each pair of students} \\
&\text{have shaken hands exactly once, and } k \text{ is as minimal as possible.} \\
&\text{Prove that } \sqrt{n} \leq k - 1 \leq 2\sqrt{n}. \\
&\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q2]} \\[20pt]
&\textbf{Q8.} \text{Given a fair } n\text{-sided die with sides } 1, 2, \ldots, n: \\
&\text{Roll the die. If you roll } n, \text{ you win. Else, roll again.} \\
&\text{HOWEVER, if your roll is not greater than your previous roll, you lose.} \\[10pt]
&\text{E.g. } n = 4: \\
&\text{134: win, } \quad 31: \text{ lose, } \quad 122: \text{ lose, } \quad 24: \text{ win.} \\
&\text{Find the probability that you win for any given } n \text{ without using summation.} \\
&\text{[Original Question]}
\end{aligned}
\]](http://latex.artofproblemsolving.com/c/f/b/cfb583f0d96a9d709a5896acc0f3f8b3938e9acc.png)
