Found problems: 85335
2012 Princeton University Math Competition, B5
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
1996 IMO Shortlist, 8
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]
2017 Bosnia And Herzegovina - Regional Olympiad, 1
If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$
2019 Purple Comet Problems, 1
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]
2004 Czech and Slovak Olympiad III A, 1
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
2018 CMIMC Team, 10-1/10-2
Find the smallest positive integer $k$ such that $ \underbrace{11\cdots 11}_{k\text{ 1's}}$ is divisible by $9999$.
Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$. Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively. If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.
2015 BMT Spring, 10
Let $ABC$ be a triangle with points $E, F$ on $CA$, $AB$, respectively. Circle $C_1$ passes through $E, F$ and is tangent to segment $BC$ at $D$. Suppose that $AE = AF = EF = 3$, $BF = 1$, and $CE = 2$. What is $\frac{ED^2}{F D^2}$ ?
1935 Moscow Mathematical Olympiad, 011
In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2007 India IMO Training Camp, 2
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$
2007 Princeton University Math Competition, 10
if $x$, $y$, and $z$ are real numbers such that $ x^2 + z^2 = 1 $ and $ y^2 + 2y \left( x + z \right) = 6 $, find the maximum value of $ y \left( z - x \right) $.
2013 MTRP Senior, 7
Write 11 numbers on a sheet of paper six zeros and five ones. Perform the following operation 10 times: cross out any two numbers, and if they were equal, write another zero on the board. If they were not equal, write a one. Show that no matter which numbers are chosen at each step, the nal number on the board will be a one.
2012 Pre - Vietnam Mathematical Olympiad, 1
For $a,b,c>0: \; abc=1$ prove that
\[a^3+b^3+c^3+6 \ge (a+b+c)^2\]
2017 JBMO Shortlist, NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.
Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
2009 National Olympiad First Round, 20
Let $ A$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is odd. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
Let $ B$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is even. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
$ A \minus{} B \equal{} ?$
$\textbf{(A)}\ 5000 \qquad\textbf{(B)}\ 4640 \qquad\textbf{(C)}\ 3200 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$
2019 IFYM, Sozopol, 4
For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and
$AB=CB$. Prove that $AD=CD$.
2005 India National Olympiad, 1
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$
than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
2009 Today's Calculation Of Integral, 461
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$.
(1) Find $ I_1,\ I_2$.
(2) Find $ \lim_{n\to\infty} I_n$.
2021 Princeton University Math Competition, A4 / B6
Let $BCDE$ be a trapezoid with $BE\parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be the intersection of $AM$ and $BE$. If $BX$ can be written as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, find $a + b$
2019 LIMIT Category B, Problem 9
The number of solutions of the equation $\tan x+\sec x=2\cos x$, where $0\le x\le\pi$, is
$\textbf{(A)}~0$
$\textbf{(B)}~1$
$\textbf{(C)}~2$
$\textbf{(D)}~3$
2023 Novosibirsk Oral Olympiad in Geometry, 3
Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.
2022 USA TSTST, 9
Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties:
[list=disc]
[*]Each element of the sequence is between $1$ and $n$, inclusive.
[*]For any two different contiguous subsequence of the sequence with length between $2$ and $k$ inclusive, the multisets of values in those two subsequences is not the same.
[*]The sequence has length at least $0.499n^2$
[/list]
2020 Online Math Open Problems, 13
Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive integers such that \[ \frac{a^2-2}{x} = \frac{b^2-37}{y} = \frac{c^2-41}{z} = a+b+c. \] Let $S=a+b+c+x+y+z$. Compute the sum of all possible values of $S$.
[i]Proposed by Luke Robitaille[/i]
2019 India IMO Training Camp, P3
Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.
2024 Kyiv City MO Round 1, Problem 2
$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$.
[i]Proposed by Bogdan Rublov[/i]