Found problems: 85335
2017 Czech-Polish-Slovak Junior Match, 2
Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.
2016 Taiwan TST Round 1, 2
Circles $O_1$ and $O_2$ intersects at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. We join $AB$ to intersect $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume that $H$ is an arbitrary point on the line segment $AF$. We join $HE$ and extend it to intersect circle $O_1$ at point $G$, and join $BG$ and extend it to intersect the extended line of $AC$ at point $D$.
Prove that $\frac{AH}{HF}=\frac{AC}{CD}$.
2018 AIME Problems, 14
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.
STEMS 2022 Math Cat A Qualifier Round, 2
Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$
Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$.
Find the last 2 digits in the decimal representation of $g(2021)$.
2012 CHMMC Fall, 10
Let
$$N = {2^{2012} \choose 0} {2^{2012} \choose 1} {2^{2012} \choose 2} {2^{2012} \choose 3}... {2^{2012} \choose 2^{2012}}.$$
Let M be the number of $0$’s when $N$ is written in binary. How many $0$’s does $M$ have when written in binary?
(Warning: this question is very hard.)
2019 MIG, 19
Let $S(n)$ denote the sum of digits of an integer $n$ (For example, $S(17) = 1 + 7 = 8$). If a positive two digit integer is randomly selected, what is the probability $S(S(n)) \ge 8$?
$\textbf{(A) }0\qquad\textbf{(B) }\dfrac19\qquad\textbf{(C) }\dfrac29\qquad\textbf{(D) }\dfrac{11}{45}\qquad\textbf{(E) }\dfrac{13}{45}$
1979 Chisinau City MO, 174
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.
2007 China National Olympiad, 2
Let $\{a_n\}_{n \geq 1}$ be a bounded sequence satisfying
\[a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots \]
Show that
\[a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots\]
1991 IMO Shortlist, 28
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2000 National High School Mathematics League, 3
There are $n$ people, any two of them have called each other at most once. In any group of $n-2$ of them, anyone of the group has called with other people in this group for $3^k$ times, where $k$ is a non-negative integer (the value of $k$ is fixed). Find all possible integers $n$.
1985 AMC 12/AHSME, 18
Six bags of marbles contain $ 18$, $ 19$, $ 21$, $ 23$, $ 25$, and $ 34$ marbles, respectively. One bag contains chipped marbles only. The other $ 5$ bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
$ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 25$
2017 Pan-African Shortlist, C2
On a $50 \times 50$ chessboard, we put, in the lower left corner, a die whose faces are numbered from $1$ to $6$. By convention, the sum of digits on two opposite side of the die equals $7$. Adama wants to move the die to the diagonally opposite corner using the following rule: at each step, Adama can roll the die only on to its right side, or to its top side. We suppose that whenever the die lands on a square, the number on its bottom face is printed on the square. By the end of these operations, Adama wants to find the sum of the $99$ numbers appearing on the chessboard. What are the maximum and minimum possible values of this sum?
2010 Korea National Olympiad, 2
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that
\[ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} \]
2018 IMC, 8
Let $\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}$. A frog moves along the points of $\Omega$ by jumps of length $1$. For every positive integer $n$, determine the number of paths the frog can take to reach $(n,n,n)$ starting from $(0,0,0)$ in exactly $3n$ jumps.
[i]Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University[/i]
Russian TST 2015, P3
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
2015 Czech-Polish-Slovak Junior Match, 5
Determine all natural numbers$ n> 1$ with the property:
For each divisor $d> 1$ of number $n$, then $d - 1$ is a divisor of $n - 1$.
Math Hour Olympiad, Grades 8-10, 2014.5
An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?
1968 AMC 12/AHSME, 4
Define an operation $*$ for positve real numbers as $a*b=\dfrac{ab}{a+b}$. Then $4*(4*4)$ equals:
$\textbf{(A)}\ \frac{3}{4} \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ \dfrac{4}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ \dfrac{16}{3} $
2017 Romania Team Selection Test, P1
a) Determine all 4-tuples $(x_0,x_1,x_2,x_3)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 4) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}$ is an interger.
b)Show that there are infinitely many 5-tuples $(x_0,x_1,x_2,x_3,x_4)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 5) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}$ is an interger.
2024 CMIMC Combinatorics and Computer Science, 2
Robert has two stacks of five cards numbered 1--5, one of which is randomly shuffled while the other is in numerical order. They pick one of the stacks at random and turn over the first three cards, seeing that they are 1, 2, and 3 respectively. What is the probability the next card is a 4?
[i]Proposed by Connor Gordon[/i]
1996 Korea National Olympiad, 5
Find all integer solution triple $(x,y,z)$ such that $x^2+y^2+z^2-2xyz=0.$
1986 All Soviet Union Mathematical Olympiad, 432
Given $30$ equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
2009 Irish Math Olympiad, 2
For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$
Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
1994 Tournament Of Towns, (401) 3
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)