Found problems: 85335
2003 National Olympiad First Round, 18
What is the least integer $n>2003$ such that $5^n + n^5$ is a multiple of $11$?
$
\textbf{(A)}\ 2010
\qquad\textbf{(B)}\ 2011
\qquad\textbf{(C)}\ 2012
\qquad\textbf{(D)}\ 2014
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2018 Regional Olympiad of Mexico Northeast, 1
$N$ different positive integers are arranged around a circle , in such a way that the sum of every $5$ consecutive numbers in the circle is a multiple of $13$. Let $A $ be the smallest possible sum of the $n$ numbers. Calculate the value of $A$ for
$\bullet$ $n = 99$,
$\bullet$ $n = 100$.
2003 National Olympiad First Round, 19
At least how many elements does the set which contains all of the midpoints of segments connecting $2003$ different points in a plane have?
$
\textbf{(A)}\ 2006
\qquad\textbf{(B)}\ 4001
\qquad\textbf{(C)}\ 4003
\qquad\textbf{(D)}\ 4006
\qquad\textbf{(E)}\ \text{None of the preceeding}
$
2001 CentroAmerican, 2
Let $ AB$ be the diameter of a circle with a center $ O$ and radius $ 1$. Let $ C$ and $ D$ be two points on the circle such that $ AC$ and $ BD$ intersect at a point $ Q$ situated inside of the circle, and $ \angle AQB\equal{} 2 \angle COD$. Let $ P$ be a point that intersects the tangents to the circle that pass through the points $ C$ and $ D$.
Determine the length of segment $ OP$.
2006 Pre-Preparation Course Examination, 3
The bell number $b_n$ is the number of ways to partition the set $\{1,2,\ldots,n\}$. For example $b_3=5$. Find a recurrence for $b_n$ and show that $b_n=e^{-1}\sum_{k\geq 0} \frac{k^n}{k!}$. Using a combinatorial proof show that the number of ways to partition $\{1,2,\ldots,n\}$, such that now two consecutive numbers are in the same block, is $b_{n-1}$.
2011 ISI B.Math Entrance Exam, 2
Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.
2004 France Team Selection Test, 1
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.
Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
Gheorghe Țițeica 2025, P4
Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$.
[i]Janez Šter[/i]
1939 Moscow Mathematical Olympiad, 048
Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients
2013 AMC 10, 4
A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
$ \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 $
1998 Harvard-MIT Mathematics Tournament, 4
Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.
2003 Italy TST, 2
Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.
2022 ISI Entrance Examination, 8
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.
2012 China Team Selection Test, 1
Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions:
[list]
[*] $a_1>a_2>\ldots>a_n$;
[*] $\gcd (a_1,a_2,\ldots,a_n)=1$;
[*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]
2018 AIME Problems, 11
Find the number of permutations of $1,2,3,4,5,6$ such that for each $k$ with $1\leq k\leq 5$, at least one of the first $k$ terms of the permutation is greater than $k$.
2016 Argentina National Olympiad Level 2, 1
In the cells of a $1 \times 100$ board, Julián writes all the integers from $1$ to $100$ (inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are $7$, $99$ and $22$, then the cell with $22$ is marked. Let $S$ be the sum of all the numbers in the marked cells. Determine the minimum value that $S$ can take.
[b]Note:[/b] Each marked number contributes to the sum $S$ exactly once, but it can be marked multiple times.
1990 ITAMO, 4
Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$. Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
1976 Euclid, 1
Source: 1976 Euclid Part A Problem 1
-----
In the diagram, $ABCD$ and $EFGH$ are similar rectangles. $DK:KC=3:2$. Then rectangle $ABCD:$ rectangle $EFGH$ is equal to
[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);
draw((55,5)--(60,5)--(60,0));
draw((75,5)--(80,5)--(80,0));
label("A",(0,50),NW);
label("B",(0,0),SW);
label("C",(75,0),SE);
label("D",(75,50),NE);
label("E",(55,20),NW);
label("F",(55,0),SW);
label("G",(100,0),SE);
label("H",(100,20),NE);
label("K",(75,20),NE);[/asy]
$\textbf{(A) } 3:2 \qquad \textbf{(B) } 9:4 \qquad \textbf{(C) } 5:2 \qquad \textbf{(D) } 25:4 \qquad \textbf{(E) } 6:2$
2013 Turkey Junior National Olympiad, 2
Find all prime numbers $p, q, r$ satisfying the equation
\[ p^4+2p+q^4+q^2=r^2+4q^3+1 \]
2004 Romania Team Selection Test, 6
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
KoMaL A Problems 2021/2022, A. 803
Let $\pi(n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\ldots, n\}$ is called [i]primitive[/i] if there are no two elements in it with one of them dividing the other. Prove that for $n\geq 5$ and $1\leq k\leq \pi(n)/2,$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements.
[i]Proposed by Cs. Sándor, Budapest[/i]
2001 Federal Math Competition of S&M, Problem 1
Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.
1996 IMO Shortlist, 4
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$.
(b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.
2013 Germany Team Selection Test, 3
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?