Found problems: 85335
2016 Kosovo Team Selection Test, 4
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied
$f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$
If $g(x)=f(x)-x+1$ then find $g(2016)$ .
2021 ASDAN Math Tournament, 2
For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2+\{\alpha\}=21$$ can be expressed in the form $$\frac{\sqrt{a}-\sqrt{b}}{c}-d$$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$
1974 IMO Shortlist, 9
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that
\[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]
1989 IMO Shortlist, 22
Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i \equal{} 1,2, \ldots, 117\}$ such that
[b]i.)[/b] each $ A_i$ contains 17 elements
[b]ii.)[/b] the sum of all the elements in each $ A_i$ is the same.
2003 Italy TST, 1
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
STEMS 2023 Math Cat A, 6
Define a positive integer $n$ to be a fake square if either $n = 1$ or $n$ can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer $k \geqslant 2$, there exist distinct positive integers $a_1$, $a_2, \cdots, a_k$ such that the polynomial $(x+a_1)(x+a_2) \cdots (x+a_k)$ takes ‘fake square’ values for all $x = 1,2,\cdots,2023$.
[i]Proposed by Prof. Aditya Karnataki[/i]
2004 AMC 12/AHSME, 9
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $ 25\%$ without altering the volume, by what percent must the height be decreased?
$ \textbf{(A)}\ 10\% \qquad \textbf{(B)}\ 25\% \qquad \textbf{(C)}\ 36\% \qquad \textbf{(D)}\ 50\% \qquad \textbf{(E)}\ 60\%$
2010 Germany Team Selection Test, 3
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
1990 AMC 12/AHSME, 10
An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\textbf{(A) }328\qquad
\textbf{(B) }329\qquad
\textbf{(C) }330\qquad
\textbf{(D) }331\qquad
\textbf{(E) }332\qquad$
2011 Mathcenter Contest + Longlist, 11
Let $a,b,c\in R^+$ with $a+b+c=3$. Prove that $$2(ab+bc+ca)\le 5+ abc$$ [i](Real Matrik)[/i]
2016 CMIMC, 1
Let \[f(x)=\dfrac{1}{1-\dfrac{1}{1-x}}\,.\] Compute $f^{2016}(2016)$, where $f$ is composed upon itself $2016$ times.
2021 Malaysia IMONST 1, 12
Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.
PEN K Problems, 3
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]
2022 Bangladesh Mathematical Olympiad, 6
About $5$ years ago, Joydip was researching on the number $2017$. He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$, and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$. Prove that $N=1008.$
1966 Czech and Slovak Olympiad III A, 3
A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.
2018 Sharygin Geometry Olympiad, 3
Let $AL$ be the bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.
2009 Tournament Of Towns, 1
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.
2014 Contests, 1
On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$.
[i]S. Berlov[/i]
2007 Germany Team Selection Test, 1
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)
[b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.
[b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
2005 Today's Calculation Of Integral, 2
Calculate the following indefinite integrals.
[1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$
[2]$\int \frac{dx}{\cos ^2 (3x+4)}$
[3]$\int (x-1)\sqrt[3]{x-2}dx$
[4]$\int x\cdot 3^{x^2+1}dx$
[5]$\int \frac{dx}{\sqrt{1-x}}dx$
2013 Tournament of Towns, 4
Eight rooks are placed on a $8\times 8$ chessboard, so that no two rooks attack one another.
All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.
2015 AMC 10, 16
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
$\textbf{(A) } \dfrac{9}{1000}
\qquad\textbf{(B) } \dfrac{1}{90}
\qquad\textbf{(C) } \dfrac{1}{80}
\qquad\textbf{(D) } \dfrac{1}{72}
\qquad\textbf{(E) } \dfrac{2}{121}
$
2019 Spain Mathematical Olympiad, 1
An integer set [i][b]T[/b][/i] is orensan if there exist integers[b] a<b<c[/b], where [b]a [/b]and [b]c[/b] are part of [i][b]T[/b][/i], but [b]b[/b] is not part of [b][i]T[/i][/b]. Count the number of subsets [b][i]T[/i][/b] of {1,2,...,2019} which are orensan.
2019 Moldova Team Selection Test, 9
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
1995 Tournament Of Towns, (448) 4
Can the number $a + b + c + d$ be prime if $a, b, c$ and $d$ are positive integers and $ab = cd$?