Found problems: 109
2021 JBMO Shortlist, C2
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]
2021 JHMT HS, 10
Let $P$ be a set of nine points in the Cartesian coordinate plane, no three of which lie on the same line. Call an ordering $\{Q_1, Q_2, \ldots, Q_9\}$ of the points in $P$ [i]special[/i] if there exists a point $C$ in the same plane such that $CQ_1 < CQ_2 < \cdots < CQ_9$. Over all possible sets $P,$ what is the largest possible number of distinct special orderings of $P?$
2022 Estonia Team Selection Test, 1
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
2021 JHMT HS, 8
For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2021 JHMT HS, 8
Find the unique integer $a > 1$ that satisfies
\[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]
2022 Azerbaijan BMO TST, G1
Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$
and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and
let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be
the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle
with center $G$ and radius $GD$.
Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.
2021 JHMT HS, 7
A line passing through $(20,21)$ intersects the curve $y = x^3-2x^2-3x+5$ at three distinct points $A, B,$ and $C,$ such that $B$ is the midpoint of $\overline{AC}$. The slope of this line is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2021 JHMT HS, 5
Terry decides to practice his arithmetic by adding the numbers between $10$ and $99$ inclusive. However, he accidentally swaps the digits of one of the numbers, and thus gets the incorrect sum of $4941.$ What is the largest possible number whose digits Terry could have swapped in the summation?
2021 Balkan MO Shortlist, G1
Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$
and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and
let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be
the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle
with center $G$ and radius $GD$.
Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.
2021 Balkan MO Shortlist, A2
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x^2 + y) \ge (\frac{1}{x} + 1)f(y)$$
holds for all $x \in \mathbb{R} \setminus \{0\}$ and all $y \in \mathbb{R}$.
2021 JBMO Shortlist, G2
Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$. If
$$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$
prove that $AP \perp BC$.
Proposed by [i]Mehmet Akif Yıldız, Turkey[/i]
2021-IMOC, C4
There is a city with many houses, where the houses are connected by some two-way roads. It is known that for any two houses $A,B$, there is exactly one house $C$ such that both $A,B$ are connected to $C$. Show that for any two houses not connected directly by a road, they have the same number of roads adjacent to them.
[i]ST[/i]
2021 Balkan MO Shortlist, C2
Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum
$$\sum_{i = 1}^{n} m_ia_i$$
is not divisible by $N$. What is the largest possible value of $n$?
[i]Proposed by Ilija Jovcevski, North Macedonia[/i]
2022 Bulgaria EGMO TST, 5
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
2021 JHMT HS, 3
Let $ABCDEF$ be a convex hexagon such that $AB=CD=EF=20, \ BC=DE=FA=21,$ and $\angle A=\angle C=\angle E=90^{\circ}.$ The area of $ABCDEF$ can then be expressed in the form $a+\tfrac{b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d.$
2021 JHMT HS, 4
Triangle $ABC$ has side lengths $AC=3, \ BC=4,$ and $AB=5.$ Let $R$ be a point on the incircle $\omega$ of $\triangle{ABC}.$ The altitude from $C$ to $\overline{AB}$ intersects $\omega$ at points $P$ and $Q.$ Then, the greatest possible area of $\triangle{PQR}$ is $\tfrac{m\sqrt n}p,$ where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$
2021 JBMO Shortlist, A3
Let $n$ be a positive integer. A finite set of integers is called $n$-divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$-divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$, or $\{1, 5, 7\}$
and $\{3, 4, 6\}$. Find all the integers $n > 0$ for which there exists a $n$-divided set.
Proposed by [i]Martin Rakovsky, France[/i]
2021 Balkan MO Shortlist, G4
Let $ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$. Let the height from $A$ cut its side
$BC$ at $D$. Let $I, I_B, I_C$ be the incenters of triangles $ABC, ABD, ACD$ respectively. Let also
$EB, EC$ be the excenters of $ABC$ with respect to vertices $B$ and $C$ respectively. If $K$ is the
point of intersection of the circumcircles of $E_CIB_I$ and $E_BIC_I$, show that $KI$ passes through
the midpoint $M$ of side $BC$.
2021 JHMT HS, 6
Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$
2021 JHMT HS, 4
Let $a_1,a_2,a_3,\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n.$ Suppose $a_1 = 1,$ $a_2 = 3,$
\[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \]
Then $c - b + \tfrac{c-b}{b}$ can be written in the form $x^y,$ where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y.$
2021 JHMT HS, 1
Dan has a fair $6$-sided die with faces labeled $1,2,3,4,+,$ and $-.$ In order to complete the equation
\[ \underline{\qquad} \ \underline{\qquad} \ \underline{\qquad}=\underline{\qquad}, \]
Dan repeatedly rolls his die and fills in a blank with the character he obtained, starting with the leftmost blank and progressing rightward. The probability that, when all blanks are filled, Dan forms a true equation, is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p+q.$
2021 JHMT HS, 3
Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$
2021 Balkan MO Shortlist, N3
Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides
every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.
2021 JBMO Shortlist, G3
Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$. The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $AE$, such that $2 \cdot FD = AE$. Let $l$ be the perpendicular to $OF$ through $F$. Prove that $l$, the tangent to $\omega$ at $E$, and the line $BC$ are concurrent.
Proposed by [i] Stefan Lozanovski, Macedonia[/i]
2021 JHMT HS, 8
Sasha has a bag that holds $6$ red marbles and $7$ green marbles. How many ways can Sasha pick a handful of (zero or more) marbles from the bag such that her handful contains at least as many red marbles as green marbles (any two marbles are distinguishable, even if they have the same color)?