Found problems: 85335
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]
2022 CCA Math Bonanza, TB3
Given that $(2\cos^2{7.5}-\cos{75}-1)^2$ can be expressed as $\frac{p}{q}$, what is $p+q$?
[i]2022 CCA Math Bonanza Tiebreaker Round #3[/i]
2024 CCA Math Bonanza, T7
Find the number of $4$ digit positive integers $n$ such that the largest power of 2 that divides $n!$ is $2^{n-1}$.
[i]Team #7[/i]
2017 Romania EGMO TST, P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1987 IMO Longlists, 50
Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$.
[hide="Variants"]Variants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$[/hide]
2004 Regional Olympiad - Republic of Srpska, 4
A convex $n$-gon $A_1A_2\dots A_n$ $(n>3)$ is divided into triangles by non-intersecting diagonals.
For every vertex the number of sides issuing from it is even, except for the vertices
$A_{i_1},A_{i_2},\dots,A_{i_k}$, where $1\leq i_1<\dots<i_k\leq n$. Prove that $k$ is even and
\[n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3\]
if $k>0$ and
\[n\equiv0\pmod3\mbox{ for }k=0.\]
Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.