Found problems: 89
2020 Brazil National Olympiad, 3
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:
[list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;
[*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list]
Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.
2020 JBMO Shortlist, 5
The positive integer $k$ and the set $A$ of distinct integers from $1$ to $3k$ inclusively are such that there are no distinct $a$, $b$, $c$ in $A$ satisfying $2b = a + c$. The numbers from $A$ in the interval $[1, k]$ will be called [i]small[/i]; those in $[k + 1, 2k]$ - [i]medium[/i] and those in $[2k + 1, 3k]$ - [i]large[/i]. It is always true that there are [b]no[/b] positive integers $x$ and $d$ such that if $x$, $x + d$, and $x + 2d$ are divided by $3k$ then the remainders belong to $A$ and those of $x$ and $x + d$ are different and are:
a) small? $\hspace{1.5px}$ b) medium? $\hspace{1.5px}$ c) large?
([i]In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$[/i].)
2020 ISI Entrance Examination, 7
Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then
[b](a)[/b] Prove that $d\leqslant 2$.
[b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .
2020 CMIMC Team, 2
Find all sets of five positive integers whose mode, mean, median, and range are all equal to $5$.
MOAA Team Rounds, TO3
Consider the addition $\begin{tabular}{cccc}
& O & N & E \\
+ & T & W & O \\
\hline
F & O & U & R \\
\end{tabular}$ where different letters represent different nonzero digits.
What is the smallest possible value of the four-digit number $FOUR$?
2020 CMIMC Combinatorics & Computer Science, 3
Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order.
The following action is performed repeatedly (any number of times):
[code]
def action():
Choose an integer n between 1 and 10 inclusive
Reverse the array between indices 1 and n inclusive
Reverse the array between indices n+1 and 10 inclusive (If n = 10, we do nothing)
[/code]
How many possible orders can the array have after we are done with this process?
2020 CMIMC Combinatorics & Computer Science, 7
Consider a complete graph of $2020$ vertices. What is the least number of edges that need to be marked such that each triangle ($3$-vertex subgraph) has an odd number of marked edges?
2020 ISI Entrance Examination, 2
Let $a$ be a fixed real number. Consider the equation
$$
(x+2)^{2}(x+7)^{2}+a=0, x \in R
$$
where $R$ is the set of real numbers. For what values of $a$, will the equ have exactly one double-root?
2020 JBMO Shortlist, 2
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$, and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \neq A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ and $(c_1)$ be the circumcircles of the triangles $\triangle AEZ$ and $\triangle BEZ$, respectively. Let $(c_2)$ be an arbitrary circle passing through the points $A$ and $E$. Suppose $(c_1)$ meets the line $CZ$ again at the point $F$, and meets $(c_2)$ again at the point $N$. If $P$ is the other point of intersection of $(c_2)$ with $AF$, prove that the points $N$, $B$, $P$ are collinear.
MOAA Team Rounds, TO4
Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$
Determine $k^2$.
2020 Indonesia MO, 2
Problem 2. Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$
2020 CMIMC Team, 3
Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.
2020 CMIMC Algebra & Number Theory, 2
Find the unique real number $c$ such that the polynomial $x^3+cx+c$ has exactly two real roots.
2020 CMIMC Algebra & Number Theory, 9
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x,y)$ such that $1\le x,y \le p$ and $x^3-3xy+y^3+1$ is divisible by $p$.
2020 JBMO Shortlist, 6
Are there any positive integers $m$ and $n$ satisfying the equation
$m^3 = 9n^4 + 170n^2 + 289$ ?
2020 Indonesia MO, 3
The wording is just ever so slightly different, however the problem is identical.
Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.
2020 CMIMC Geometry, 9
In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.
2020 JBMO Shortlist, 2
Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and
$a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$
for $n \ge 1$.
Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.
2020 MOAA, TO3
Consider the addition $\begin{tabular}{cccc}
& O & N & E \\
+ & T & W & O \\
\hline
F & O & U & R \\
\end{tabular}$ where different letters represent different nonzero digits.
What is the smallest possible value of the four-digit number $FOUR$?
2020 CMIMC Combinatorics & Computer Science, 2
David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly?
2020 CMIMC Team, 14
Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?
2020 CMIMC Team, 15
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?
2020 JBMO Shortlist, 7
Prove that there doesn’t exist any prime $p$ such that every power of $p$ is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).
2020 CMIMC Team, 4
Given $n=2020$, sort the $6$ values $$n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}$$ from [b]least[/b] to [b]greatest[/b]. Give your answer as a 6 digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
2020 CMIMC Geometry, 5
For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that
$$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$ What is $x+y$?
(A [i]homothety[/i] $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)