Found problems: 85335
2024 Chile TST Ibero., 3
Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:
1997 Finnish National High School Mathematics Competition, 2
Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn.
This tangent and the circles bound a region inside which a circle as large as possible is drawn.
What is the radius of this circle?
1990 AMC 8, 4
Which of the following could not be the unit's digit [one's digit] of the square of a whole number?
$ \text{(A)}\ 1\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 8 $
2019 China Team Selection Test, 4
Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?
Here $A+B=\{a+b|a\in A, b\in B\}$.
2014 Math Prize For Girls Problems, 18
For how many integers $k$ such that $0 \le k \le 2014$ is it true that the binomial coefficient $\binom{2014}{k}$ is a multiple of 4?
2020 LMT Fall, B3
Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.
2024 Korea National Olympiad, 8
On a blackboard, there are $10$ numbers written: $1, 2, \dots, 10$. Nahyun repeatedly performs the following operations.
[b](Operation)[/b] Nahyun chooses two numbers from the 10 numbers on the blackboard that are not in a divisor-multiple relationship, erases them, and writes their GCD and LCM on the blackboard.
If every two numbers on the blackboard form a divisor-multiple relationship, Nahyun stops the process. What is the maximum number of operations Nahyun can perform?
(Note: $a, b$ are in a divisor-multiple relationship iff $a \mid b$ or $b \mid a$.)
2018 Thailand TST, 2
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
2022 Kosovo National Mathematical Olympiad, 1
$22$ light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.
1996 Canadian Open Math Challenge, 1
The roots of the equation $x^2+4x-5 = 0$ are also the roots of the equation $2x^3+9x^2-6x-5 = 0$. What is the third root of the second equation?
1986 IMO Longlists, 77
Find all integers $x,y,z$ such that
\[x^3+y^3+z^3=x+y+z=8\]
1993 Tournament Of Towns, (384) 2
The square $ PQRS$ is placed inside the square $ABCD$ in such a way that the segments $AP$, $BQ$, $CR$ and $DS$ intersect neither each other nor the square $PQRS$. Prove that the sum of areas of quadrilaterals $ABQP$ and $CDSR$ is equal to the sum of the areas of quadrilaterals $BCRQ$ and $DAPS$.
(Folklore)
2010 AIME Problems, 14
For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$.
[b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.
2008 National Olympiad First Round, 36
There is a white table with a pile of $2008$ coins and there are two empty black tables. At each move, the uppermost coin on a table is transferred to an empty table or to the top of the pile on a non-empty table. What is the least number of moves required to reverse the pile at the beginning on the white table?
$
\textbf{(A)}\ 6016
\qquad\textbf{(B)}\ 6017
\qquad\textbf{(C)}\ 6022
\qquad\textbf{(D)}\ 6023
\qquad\textbf{(E)}\ 6024
$
2023 Romania National Olympiad, 1
We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$.
a) Show that the equation has $2$ distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$
2008 Princeton University Math Competition, B7
In this problem, we consider only polynomials with integer coeffients. Call two polynomials $p$ and $q$ [i]really close[/i] if $p(2k + 1) \equiv q(2k + 1)$ (mod $210$) for all $k \in Z^+$. Call a polynomial $p$ [i]partial credit[/i] if no polynomial of lesser degree is [i]really close[/i] to it. What is the maximum possible degree of [i]partial credit[/i]?
2012 Kyrgyzstan National Olympiad, 1
Prove that $ n $ must be prime in order to have only one solution to the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{n}$, $x,y\in\mathbb{N}$.
2018 Singapore Senior Math Olympiad, 4
Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.
1985 All Soviet Union Mathematical Olympiad, 411
The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?
2024 Moldova Team Selection Test, 4
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2009 Moldova Team Selection Test, 4
let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality:
$ \dfrac{60a^2\minus{}1}{4xy\plus{}5z}\plus{}\dfrac{60b^2\minus{}1}{4yz\plus{}5x}\plus{}\dfrac{60c^2\minus{}1}{4zx\plus{}5y}\geq 12$
2018 Purple Comet Problems, 25
If a and b are in the interval $\left(0, \frac{\pi}{2}\right)$ such that $13(\sin a + \sin b) + 43(\cos a + \cos b) = 2\sqrt{2018}$, then $\tan a + \tan b = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 Kurschak Competition, 1
Consider a company of $n\ge 4$ people, where everyone knows at least one other person, but everyone knows at most $n-2$ of the others. Prove that we can sit four of these people at a round table such that all four of them know exactly one of their two neighbors. (Knowledge is mutual.)
2019 China Team Selection Test, 1
Cyclic quadrilateral $ABCD$ has circumcircle $(O)$. Points $M$ and $N$ are the midpoints of $BC$ and $CD$, and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$. Let $EN$ meet $FM$ at $P$. Denote $S$ as the circumcenter of $\triangle PEF$. Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$.
2018 Saudi Arabia IMO TST, 1
Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.