1
An acute triangle $ABC$ has $BC$ as its longest side. Points $D,E$ respectively lie on $AC,AB$ such that $BA = BD$ and $CA = CE$ . The point $A'$ is the reflection of $A$ against line $BC$ . Prove that the circumcircles of $ABC$ and $A'DE$ have the same radii.
2
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$ :
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \] Note: $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$ .
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \] Note: $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$ .
3
A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$ . Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$ , and replace $X$ with $X+d$ if Neneng chose the sign up or $X-d$ if Neneng chose down. This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero.
Prove that if $n \geq 14$ , Asep can win in at most $(n-5)/4$ steps.
Prove that if $n \geq 14$ , Asep can win in at most $(n-5)/4$ steps.
4
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
1. $N$ is divisible by $2^{2023}$ , but not by $2^{2024}$ ,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
1. $N$ is divisible by $2^{2023}$ , but not by $2^{2024}$ ,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
Day 2
30 August 2023
5
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$ . If $b \le a$ , show that $b$ is a perfect square.
6
Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$ , there exists an integer $r$ with $0 \le r \le n - k$ which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]
7
Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$ . Let $\omega$ be the circumcircle of triangle $ABC$ . The tangents of $\omega$ at $B$ and $C$ intersect at $P$ . Let $M$ be the midpoint of $PB$ . Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$ . Point $D$ is on $CM$ such that $ED \parallel BM$ . Show that the circumcircle of $CDE$ is tangent to $\omega$ .
8
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$ . For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$ , the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$ , and for all the other permutations of $(1, 2, 3)$ , the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$ .
Determine the maximum number of elements in $S(a, b, c)$ .
source : artofproblemsolving.com
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