Found problems: 109
2022 Romania Team Selection Test, 3
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.
[i]Proposed by Nikola Velov, North Macedonia[/i]
2021 JHMT HS, 8
Each of the $9$ cells in a $3\times 3$ grid is colored either blue or white with equal probability. The expected value of the area of the largest square of blue cells contained within the grid is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 Balkan MO Shortlist, C3
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
2021 JHMT HS, 7
Triangle $JHT$ has side lengths $JH = 14$, $HT = 10$, and $TJ = 16$. Points $I$ and $U$ lie on $\overline{JH}$ and $\overline{JT},$ respectively, so that $HI = TU = 1.$ Let $M$ and $N$ be the midpoints of $\overline{HT}$ and $\overline{IU},$ respectively. Line $MN$ intersects another side of $\triangle JHT$ at a point $P$ other than $M.$ Compute $MP^2.$
2021 JHMT HS, 5
The average of all ten-digit base-ten positive integers $\underline{d_9} \ \underline{d_8} \ldots \underline{d_1} \ \underline{d_0}$ that satisfy the property $|d_i - i| \leq 1$ for all $i \in \{0, 1, \ldots, 9\}$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Compute the remainder when $p + q$ is divided by $10^6.$
2021 JHMT HS, 2
David has some pennies. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $3$ pennies left; is David spends all his money on cranberries, he will have $2$ pennies left. What is the least possible amount of pennies that David can originally have?
2021 JHMT HS, 1
In the diagram below, a triangular array of three congruent squares is configured such that the top row has one square and the bottom row has two squares. The top square lies on the two squares immediately below it. Suppose that the area of the triangle whose vertices are the centers of the three squares is $100.$ Find the area of one of the squares.
[asy]
unitsize(1.25cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0));
draw((1,0)--(2,0)--(2,1)--(1,1));
draw((1.5,1)--(1.5,2)--(0.5,2)--(0.5,1));
draw((0.5,0.5)--(1.5,0.5)--(1,1.5)--(0.5,0.5),dashed);
[/asy]
2022 Azerbaijan JBMO TST, C5?
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by [i]Nikola Velov, Macedonia[/i]
2021 JHMT HS, 4
For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute
\[ \sum_{n=46}^{86} f(n). \]
2021 JBMO Shortlist, C3
We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a
triple of positive integers $(b; y; r) \in \{1; 2; ...; 7\}^3$ and flip all the jars whose number of blue, yellow and red marbles differ by not more than $1$ from $b, y, r$, respectively. After $n$ moves all the jars turned out to be with the caps down. Find the number of all possible values of $n$, if $n \le 2021$.
2021 Balkan MO Shortlist, G8
Let $ABC$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $BC, CA$,
and $AB$ are $D, E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $AI$, and let
$L$ be the second point of intersection of the circumcircles of the triangles $BFK$ and $CEK$. If
$\frac{1}{3} BC = AC - AB$, prove that $DE = 2KL$.
2021 JBMO Shortlist, C6
Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$
and $n$ where $m, n \ge 3$.
Proposed by [i]Toghrul Abbasov, Azerbaijan[/i]
2021 Balkan MO Shortlist, G6
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.
[i]Proposed by Nikola Velov, North Macedonia[/i]
2021 JBMO Shortlist, N6
Given a positive integer $n \ge 2$, we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$. For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$. Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$.
2022 Germany 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, 5
For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$
2022 Azerbaijan BMO TST, N4*
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.
2021 JHMT HS, 7
At a prom, there are $4$ boys and $3$ girls. Each boy picks a girl to dance with, and each girl picks a boy to dance with. Assuming that each choice is uniformly random, the probability that at least one boy and one girl choose each other as dance partners is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Compute $p+q.$
2021 JBMO Shortlist, A2
Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for
which the following property holds:
If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$.
Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]
2021 IMO Shortlist, A1
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, 1
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Find the value of the sum
\[ \left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor+\left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor+\cdots+\left\lfloor 2+\frac{1}{2^1} \right\rfloor+\left\lfloor 2+\frac{1}{2^0} \right\rfloor. \]
2021 Balkan MO Shortlist, C1
Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple.
Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.
2021 JHMT HS, 6
A sequence of positive integers $a_0, a_1, a_2, \dots$ satisfies $a_0 = 83$ and $a_n = (a_{n-1})^{a_{n-1}}$ for all positive integers $n$. Compute the remainder when $a_{2021}$ is divided by $60$.
2021 Balkan MO Shortlist, A5
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$
holds for all $x, y \in \mathbb{R}^{+}$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
2021 Balkan MO Shortlist, G7
Let $ABC$ be an acute scalene triangle. Its $C$-excircle tangent to the segment $AB$ meets
$AB$ at point $M$ and the extension of $BC$ beyond $B$ at point $N$. Analogously, its $B$-excircle
tangent to the segment $AC$ meets $AC$ at point $P$ and the extension of $BC$ beyond $C$ at point
$Q$. Denote by $A_1$ the intersection point of the lines $MN$ and $PQ$, and let $A_2$ be defined as the
point, symmetric to $A$ with respect to $A_1$. Define the points $B_2$ and $C_2$, analogously. Prove
that $\triangle ABC$ is similar to $\triangle A_2B_2C_2$.