Found problems: 85335
2020 Stars of Mathematics, 2
Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$
[i]The Problem Selection Committee[/i]
2010 BAMO, 2
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
2008 AMC 10, 1
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$ \textbf{(A)}$ 1:50 PM $ \qquad
\textbf{(B)}$ 3:00 PM $ \qquad
\textbf{(C)}$ 3:30 PM $ \qquad
\textbf{(D)}$ 4:30 PM $ \qquad
\textbf{(E)}$ 5:50 PM
2000 Brazil Team Selection Test, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
1987 Traian Lălescu, 2.1
For any nonegative real $ a $ and natural $ n, $ prove that
$$ \sqrt{a+1+\sqrt{a+2+\cdots +\sqrt{a+n}}} <a+3. $$
2013 India PRMO, 14
Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?
2005 Kyiv Mathematical Festival, 4
Let $ M$ be the intersection point of medians of a triangle $ \triangle ABC.$ It is known that $ AC \equal{} 2BC$ and $ \angle ACM \equal{} \angle CBM.$ Find $ \angle ACB.$
2017 Greece National Olympiad, 3
Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number
$$N=2017-a^3b-b^3c-c^3a$$ is a perfect square of an integer.
2018 Costa Rica - Final Round, 2
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.
2005 AMC 10, 7
Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
$ \textbf{(A)}\ 4\qquad
\textbf{(B)}\ 5\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 7\qquad
\textbf{(E)}\ 8$
2010 Laurențiu Panaitopol, Tulcea, 2
Let be a nonnegative integer $ n $ such that $ \sqrt n $ is not integer. Show that the function
$$ f:\{ a+b\sqrt n | a,b\in\{ 0\}\cup\mathbb{N} , a^2-nb^2=1 \}\longrightarrow\{ 0\}\cup\mathbb{N} , f(x) =\lfloor x \rfloor $$
is injective and non-surjective.
2017 Regional Competition For Advanced Students, 4
Determine all integers $n \geq 2$, satisfying
$$n=a^2+b^2,$$
where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$.
[i]Proposed by Walther Janous[/i]
2010 Indonesia TST, 3
In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party?
[i]Yudi Satria, Jakarta[/i]
1987 Spain Mathematical Olympiad, 6
For all natural numbers $n$, consider the polynomial $P_n(x) = x^{n+2}-2x+1$.
(a) Show that the equation $P_n(x)=0$ has exactly one root $c_n$ in the open interval $(0,1)$.
(b) Find $lim_{n \to \infty}c_n$.
2020 Thailand TSTST, 4
Does there exist a set $S$ of positive integers satisfying the following conditions?
$\text{(i)}$ $S$ contains $2020$ distinct elements;
$\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and
$\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b}
ab$ is not a prime power.
2019 Malaysia National Olympiad, 3
A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?
1957 Moscow Mathematical Olympiad, 367
Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.
1985 All Soviet Union Mathematical Olympiad, 404
The convex pentagon $ABCDE$ was drawn in the plane.
$A_1$ was symmetric to $A$ with respect to $B$.
$B_1$ was symmetric to $B$ with respect to $C$.
$C_1$ was symmetric to $C$ with respect to $D$.
$D_1$ was symmetric to $D$ with respect to $E$.
$E_1$ was symmetric to $E$ with respect to $A$.
How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?
2008 AMC 10, 25
Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$
2005 Bulgaria Team Selection Test, 2
Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$
2010 LMT, 16
Al has three bags, each with three marbles each. Bag $1$ has two blue marbles and one red marble, Bag $2$ has one blue marble and two red marbles, and Bag $3$ has three red marbles. He chooses two distinct bags at random, then one marble at random from each of the chosen bags. What is the probability that he chooses two blue marbles?
2011 IFYM, Sozopol, 1
Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.
2009 Romania National Olympiad, 1
On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $
2016 AMC 10, 20
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?
$\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$
1981 National High School Mathematics League, 9
$O$ is a circle with a radius of $1$, with strings $CD$ and $EF$. $CD//EF$, and diameter $AB$ intersects $CD,EF$ at $P,Q$. If $\angle BPD=\frac{\pi}{4}$, prove that
$$PC\cdot QE+PD \cdot QF<2.$$