Found problems: 85335
2016 India IMO Training Camp, 1
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
2021 HMNT, 2
Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:
$x\,\,\,\, z = 15$
$x\,\,\,\, y = 12$
$x\,\,\,\, x = 36$
If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x+10y+z$.
2008 Germany Team Selection Test, 1
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that
\[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}.
\]
[i]Author: Marcin Kuzma, Poland[/i]
2007 Tuymaada Olympiad, 3
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
2017 Rioplatense Mathematical Olympiad, Level 3, 3
Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that
$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.
2017 VTRMC, 2
Evaluate $ \int _ { 0 } ^ { a } d x / ( 1 + \cos x + \sin x ) $ for $ - \pi / 2 < a < \pi $. Use your answer to show that $ \int _ { 0 } ^ { \pi / 2 } d x / ( 1 + \cos x + \sin x ) = \ln 2 $.
1966 Swedish Mathematical Competition, 5
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.
Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.
Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?
1983 AIME Problems, 9
Find the minimum value of
\[\frac{9x^2 \sin^2 x + 4}{x \sin x}\]
for $0 < x < \pi$.
2017 CMIMC Geometry, 6
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$, $\angle ADB=50^\circ$, and $BC=CD$. Suppose $AB$ intersects $CD$ at point $P$, while $AD$ intersects $BC$ at point $Q$. Compute $\angle APQ-\angle AQP$.
1994 Mexico National Olympiad, 1
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.
2021 China Second Round Olympiad, Problem 4
When the expression $$(xy-5x+3y-15)^n$$ for some positive integer $n$ is expanded and like terms are combined, the expansion contains at least 2021 distinct terms. Compute the minimum possible value of $n$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 4)[/i]
2005 MOP Homework, 6
Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.
2016 Belarus Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2021 MIG, 1
What is $20 - 2^1$?
$\textbf{(A) }1\qquad\textbf{(B) }18\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2021 Latvia Baltic Way TST, P13
Does there exist a natural number $a$ so that:
a) $\Big ((a^2-3)^3+1\Big) ^a-1$ is a perfect square?
b) $\Big ((a^2-3)^3+1\Big) ^{a+1}-1$ is a perfect square?
2005 Regional Competition For Advanced Students, 1
Show for all integers $ n \ge 2005$ the following chaine of inequalities:
$ (n\plus{}830)^{2005}<n(n\plus{}1)\dots(n\plus{}2004)<(n\plus{}1002)^{2005}$
2017 Saudi Arabia BMO TST, 1
Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers
2015 May Olympiad, 4
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2023 MMATHS, 11
A knight is on an infinite chessboard. After exactly $100$ legal moves, how many different possible squares can it end on? A knight can move to any of the $8$ closest squares not on the same row, column, or diagonal, as illustrated in the figure below.
[center][img]https://cdn.artofproblemsolving.com/attachments/0/7/144226144fb3ead533e7b517f5f65d8a70da5a.png[/img]
[/center]
1993 Baltic Way, 10
Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be two finite sequences consisting of $2n$ real different numbers. Rearranging each of the sequences in increasing order we obtain $a_1',a_2',\ldots,a_n'$ and $b_1',b_2',\ldots,b_n'$. Prove that
\[\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.\]
2017 BMT Spring, 5
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?
2014 Canada National Olympiad, 4
The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.
2001 IMO, 5
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
2023 UMD Math Competition Part I, #5
You shoot an arrow in the air. It falls to earth, you know not where. But you do know that the arrow’s height in feet after ${t}$ seconds is $-16t^2 + 80t + 96.$ After how many seconds does the arrow hit the ground?
(the ground has height 0)
$$
\mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6
$$