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JBMO 2024  |  SHORTLIST

 

A1
Let $a, b, c$ be positive real numbers such that


$$a^2 + b^2 + c^2 = \frac{1}{4}.$$

Prove that


$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$
 
A2
Let a, b, c be real numbers such that a + b + c = 0 and abc = -16. Find the minimum value of the expression

\[ W = \frac{a^2 + b^2}{c} + \frac{b^2 + c^2}{a} + \frac{c^2 + a^2}{b}. \]
 
 
A3
Anna and Bob are constructing quadratic polynomials $f_A$ and $f_B$ as follows: With Anna starting first, they take alternate turns in choosing one by one the coefficients of the polynomials, with Anna choosing the coefficients of $f_A$, and Bob the coefficients of $f_B$. In their turn, each player can choose whichever coefficient of their polynomial is not yet chosen, with the only restriction being that the coefficients have to be positive real numbers.
Bob wins if any of the following two cases occurs:
\begin{align*} \text{(a)}&\text{ The roots of }{f_A(x)}\text{ are not real.}\\ \text{(b)}&\text{ The roots of both polynomials are real numbers and furthermore each root of } f_B(x) \text{ is strictly} \\ &\text{larger than each root of } f_A(x) \text{.} \end{align*}
Otherwise Anna wins. Determine which player has a winning strategy.
 
A4
If $a,b,c,d$ are positive real numbers such that $(a+c)^2=4(ad+bc)$ then prove that

$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}+\frac{4bd}{ac}\geq 6.$$
When does equality hold?
 
A5
Find all triples $(a, b, c)$ of positive real numbers that satisfy the system of equations

\[ a + b + c = \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}, \quad ab + bc + ca = \sqrt{a} + \sqrt{b} + \sqrt{c}. \]
 
A6
Consider the function \( f : \mathbb{R} \setminus \{-1\} \to \mathbb{R} \) defined by
$$ f(x) = \frac{2x - 1}{(1012x + 1)(1 - x)} $$
Compute

$$ \lfloor\sum_{k=0}^{2023} f(\frac{k}{2024})\rfloor $$
 
A7
Show that for any real number $k>2$, the inequality
$$\lfloor\frac{1+\sqrt{4n+1}}{2}\rfloor\leq k\sqrt{n}$$
holds for infinitely many positive integers $n$.
 
C1
Determine the smallest positive integer $k$ with the following property. For any subset $~\mathbb{S}$ of the set $\{1,2,3,\cdot,2024\}$ with $|S|=k$, there are two distinct elements $a,b\in\mathbb{S}$ such that $ab+1$ is a perfect square
 
C2
Let $n$ be a positive integer such that $n^2-1$ is divisible by $6$. An $n\times n$ board is given. Prove that it is possible to place $\frac{n^2-1}{6}$ non-overlapping right triangles on the board with the lengths of $3,4,5$.
 
C3
A set of positive integers is called arithmetic if it contains three distinc elements which form an arithmetic sequence. Prove that at least $51\%$ of the subsets of the set $\{1,2,3,\cdot ,2024\}$ are arithmetic.
 
C4
All the unit cubes of a $5\times 5\times 3$ is colored white initially. We call two unit cubes are adjacent to each other if they share a common face. What is the largest number of unit cubes we can color in black such that each black unit cube is adjacent to at most one other black unit cube?
 
G1
Let $\triangle ABC$ be an acute-angled triangle with $AB = BC$. The perpendicular bisector of $AB$ meets $BC$ at $D$, and the circumcircle $\omega$ of $\triangle ADC$ again at the point $E$. Let $F$ be diametrically opposite point of $E$ in $\omega$. Prove that $BD=DF$
 
G2
Let $ABC$ be an acute-angled triangle, and $A'$ and $B'$ be the feet of the altitudes from $A$ and $B$ to $BC$, $CA$ respectively. Let $K$ and $L$ be the reflections of $A'$ with respect to $AB$ and $AC$, respectively. Let $M$ and $N$ be the reflection of $B'$ with respect to $AB$ and $BC$, respectively. Prove that $KL = MN.$
 
G3
Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear.

(The excircle of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)
 
G4
Let $ABCD$ be a circumscribed quadrilateral with circumcircle $\omega$ such that $AE = EC$, where $E$ is the intersection point of the diagonals $AC$ and $BD$. Point $F$ is taken on $\omega$ such that $BF\parallel AC$. If $G$ is the reflection of $F$ with respect to $A$, prove that the circumcircle of $\triangle ADG$ is tangent to the line $AC$
 
G5
Let \( ABCD \) be a rectangle, and \( H \) be the midpoint of the side \( AB \). Point \( K \) is taken on \( DH \) such that \( \angle BKD = 90^\circ \). Let \( F \) be a point on the diagonal \( AC \). The perpendicular line to \( AB \) through \( F \) meets \( AB \) at \( G \), and the parallel line to \( AB \) through \( F \) meets \( DH \) at \( L \). If \( M \) is the midpoint of \( GB \), prove that the angles \( \angle AKF \) and \( \angle LKM \) are equal.
 
G6
Let \( ABCD \) be a trapezoid with \( AB \parallel CD \). Let \( E \) and \( F \) be points on \( CD \) such that \( AE \perp CD \) and \( AF \perp AD \). Let \( G \) be a point on \( AE \) such that \( BG \perp AD \). Prove that the perpendicular line from \( A \) to \( BD \) bisects the segment \( FG \).
 
G7
Let \( ABC \) be an acute-angled and scalene triangle, and \( D \) be a point on the side \( BC \). Points \( E \) and \( F \) are taken on \( AD \) such that \( EB \perp AB \) and \( FC \perp AC \). Points \( S \) and \( T \) are taken on \( BC \) such that \( SE \parallel AC \) and \( TF \parallel AB \). The circumcircle of \( \triangle BSE \) intersects \( AB \) for the second time at \( M \), and the circumcircle of \( \triangle CTF \) intersects \( AC \) for the second time at \( N \). Prove that the lines \( MS \), \( NT \), and \( AD \) are concurrent
 
G8
Let \( ABC \) be a scalene triangle with smallest side \( BC \), and \( D \) be a point on the side \( BC \) such that \( \angle CAD = \angle DAB \). Let \( \omega_1 \) and \( \omega_2 \) be the circumcircles of \( \triangle ABD \) and \( \triangle ACD \) respectively. The line \( AC \) meets \( \omega_1 \) again at \( F \), and the line \( AB \) meets \( \omega_2 \) again at \( E \). The line \( DE \) meets \( \omega_1 \) again at \( G \), and the line \( DF \) meets \( \omega_2 \) again at \( H \). Prove that circumcircles of \( \triangle ABC \), \( \triangle AEF \) and \( \triangle AGH \) have a common point other than \( A \).
 
N1
Find all pairs of positive integers $(m,n)$ such that $|4^m-7^n|$ is a prime number.
 
N2
Find all the pairs of $(p;q)$ distinct prime numbers such that such that
$$\frac{p+p^q+p^{q^p}}{q^p}$$
is a perfect square.
 
N3
Let $c\in\mathbb{Z}^+$ be a $\text{Turkish number}$ if there is a positive integer $m$ such that $m^3-m=c!$ and $m^2-1$ has less than $12$ positive divisors. Determine all $\text{Turkish numbers}$.
 
N4
For any positive integer $n$, let $s(n) = 1 + 2 + \cdots + n.$ Define a strictly increasing sequence of positive integers $\{a_n\}_{n \geq 1}$ such that $a_1 = 1$ and $ a_{n+1} = \min \left\{ m \mid s(m) - s(a_n) \text{ is a perfect square} \right\} $for all positive integers $n.$ Find the value of $a_{2024}.$
 
N5
Does there exist a positive integer k such that the number of quadruples $(a, b, c, n)$ of positive integers satisfying

\[ a^5 + b^6 + c^{15} - n! = k \]
is finite?
 
N6
Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$$2020^x + 2^y = 2024^z.$$