Found problems: 85335
2007 USA Team Selection Test, 6
For a polynomial $ P(x)$ with integer coefficients, $ r(2i \minus{} 1)$ (for $ i \equal{} 1,2,3,\ldots,512$) is the remainder obtained when $ P(2i \minus{} 1)$ is divided by $ 1024$. The sequence
\[ (r(1),r(3),\ldots,r(1023))
\]
is called the [i]remainder sequence[/i] of $ P(x)$. A remainder sequence is called [i]complete[/i] if it is a permutation of $ (1,3,5,\ldots,1023)$. Prove that there are no more than $ 2^{35}$ different complete remainder sequences.
2018 Estonia Team Selection Test, 3
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
1979 Vietnam National Olympiad, 1
Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$
2023 Stars of Mathematics, 1
A convex polygon is dissected into a finite number of triangles with disjoint interiors, whose sides have odd integer lengths. The triangles may have multiple vertices on the boundary of the polygon and their sides may overlap partially.
[list=a]
[*]Prove that the polygon's perimeter is an integer which has the same parity as the number of triangles in the dissection.
[*]Determine whether part a) holds if the polygon is not convex.
[/list]
[i]Proposed by Marius Cavachi[/i]
[i]Note: the junior version only included part a), with an arbitrary triangle instead of a polygon.[/i]
1986 Polish MO Finals, 1
A square of side $1$ is covered with $m^2$ rectangles.
Show that there is a rectangle with perimeter at least $\frac{4}{m}$.
1992 India National Olympiad, 4
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
2018-IMOC, A2
For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that
$$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.
2011 China Team Selection Test, 3
For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.
1992 Tournament Of Towns, (345) 3
Do there exist two polynomials $P(x)$ and $Q(x)$ with integer coefficients such that
$$(P-Q)(x), \,\,\,\, P(x) \,\,\,\, and \,\,\,\,(P+Q)(x)$$
are squares of polynomials (and $Q$ is not equal to $cP$, where $c$ is a real number)?
(V Prasolov)
2022 Puerto Rico Team Selection Test, 2
Suppose $a$ is a non-zero real number such that $a +\frac{1}{a}$ is a whole number.
(a) Prove that $a^2 +\frac{1}{a^2}$ is also an integer.
(b) Prove that $a^n+\frac{1}{a^n}$ is also an integer, for any integer value positive of $n$.
2022 Girls in Math at Yale, 7
Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$, inclusive, and that it is divisible by $99$, find the maximum possible value of $\overline{ABCDEF}$.
[i]Proposed by Andrew Milas[/i]
2017 NIMO Problems, 3
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i]
2008 Serbia National Math Olympiad, 1
Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$
2016 Sharygin Geometry Olympiad, P19
Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.
2008 Hanoi Open Mathematics Competitions, 1
How many integers are there in $(b,2008b]$, where $b$ ($b > 0$) is given.
2018 Germany Team Selection Test, 2
A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.
1970 AMC 12/AHSME, 10
Let $F=.48181\cdots$ be an infinite repeating decimal with the digits $8$ and $1$ repeating. When $F$ is written as a fraction in lowest terms, the denominator exceeds the numerator by
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }29\qquad\textbf{(D) }57\qquad \textbf{(E) }126$
2005 District Olympiad, 3
a)Let $A,B\in \mathcal{M}_3(\mathbb{R})$ such that $\text{rank}\ A>\text{rank}\ B$. Prove that $\text{rank}\ A^2\ge \text{rank}\ B^2$.
b)Find the non-constant polynomials $f\in \mathbb{R}[X]$ such that $(\forall)A,B\in \mathcal{M}_4(\mathbb{R})$ with $\text{rank}\ A>\text{rank}\ B$, we have $\text{rank}\ f(A)>\text{rank}\ f(B)$.
1998 Romania Team Selection Test, 1
Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that:
i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$
ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$
[i]Radu Todor[/i]
2024 Junior Balkan Team Selection Tests - Romania, P4
Let $ABC$ be a triangle. An arbitrary circle which passes through the points $B,C$ intersects the sides $AC,AB$ for the second time in $D,E$ respectively. The line $BD$ intersects the circumcircle of the triangle $AEC$ at $P{}$ and $Q{}$ and the line $CE$ intersects the circumcircle of the triangle $ABD$ at $R{}$ and $S{}$ such that $P{}$ is situated on the segment $BD{}$ and $R{}$ lies on the segment $CE.$ Prove that:
[list=a]
[*]The points $P,Q,R$ and $S{}$ are concyclic.
[*]The triangle $APQ$ is isosceles.
[/list]
[i]Petru Braica[/i]
2019 CCA Math Bonanza, L3.1
Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$. What is the minimum possible value of $N$?
[i]2019 CCA Math Bonanza Lightning Round #3.1[/i]
Novosibirsk Oral Geo Oly VIII, 2023.4
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.
2014 Stanford Mathematics Tournament, 9
In cyclic quadrilateral $ABCD$, $AB= AD$. If $AC = 6$ and $\frac{AB}{BD} =\frac35$ , find the maximum possible area of $ABCD$.
2020 AMC 10, 3
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?
$\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $
2012 Today's Calculation Of Integral, 774
Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$