Found problems: 85335
2023 Euler Olympiad, Round 2, 1
Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$?
[i]Proposed by Nika Glunchadze, Georgia[/i]
2012 Germany Team Selection Test, 2
Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$
2006 IMO Shortlist, 6
Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$.
Find all local champions and determine their number.
[i]Proposed by Zoran Sunic, USA[/i]
2024 Malaysian IMO Training Camp, 5
Do there exist infinitely many positive integers $a, b$ such that $$(a^2+1)(b^2+1)((a+b)^2+1)$$ is a perfect square?
[i]Proposed Ivan Chan Guan Yu[/i]
2008 Oral Moscow Geometry Olympiad, 5
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov).
1993 All-Russian Olympiad Regional Round, 11.6
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
2013 Harvard-MIT Mathematics Tournament, 18
Define the sequence of positive integers $\{a_n\}$ as follows. Let $a_1=1$, $a_2=3$, and for each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$). For example, $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$. Express $a_{2013}$ in base $10$.
1982 Swedish Mathematical Competition, 1
How many solutions does
\[
x^2 - [x^2] = \left(x - [x]\right)^2
\]
have satisfying $1 \leq x \leq n$?
2021 Sharygin Geometry Olympiad, 10-11.8
On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_2$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.
TNO 2008 Junior, 8
A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements:
**Person 1:** The red car was guilty, and either the green or the blue one was involved.
**Person 2:** Either the green car or the red car was guilty, but not both.
**Person 3:** Only one of the cars was guilty, but it was not the blue one.
The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied.
Which car(s) were responsible for the accident?
2014 Costa Rica - Final Round, 4
The Olcommunity consists of the next seven people: Christopher Took, Humberto Brandybuck, German son of Isildur, Leogolas, Argimli, Samzamora and Shago Baggins. This community needs to travel from the Olcomashire to Olcomordor to save the world. Each person can take with them a total of $4$ day-provisions, that can be transferred to other people that are on the same day of traveling, as long as nobody holds more than $4$ day-provisions at the time. If a person returns to Olcomashire, they will be too tired to go out again. What is the farthest away Olcomordor can be from Olcomashire, so that Shago Baggins can get to Olcomordor, and the rest of the Olcommunity can return save to Olcomashire?
Note: All the members of the Olcommunity should eat exactly one day-provision while they are away from Olcomashire. The members only travel an integer number of days on each direction. Members of the Olcommunity may leave Olcomashire on different days.
2008 AMC 12/AHSME, 18
Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$?
$ \textbf{(A)}\ \sqrt{85} \qquad
\textbf{(B)}\ \sqrt{90} \qquad
\textbf{(C)}\ \sqrt{95} \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ \sqrt{105}$
1992 Taiwan National Olympiad, 4
For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.
1991 IMO Shortlist, 23
Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that
[i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$
[i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$
2013 Putnam, 6
Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either
• places a stone in an empty space, or
• removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists).
Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?
2015 USAMTS Problems, 5
Let $a_1,a_2,\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$, $a_2=-1$ and the remaining numbers are $0$. After every second, we perform the following process on the sequence: for $i=1,2,\dots,99$, replace $a_i$ with $a_i+a_{i+1}$, and replace $a_{100}$ with $a_{100}+a_1$. (All of this is done simultaneously, so each new term is the sum of two terms of the sequence from before any replacements.) Show that for any integer $M$, there is some index $i$ and some time $t$ for which $|a_i|>M$ at time $t$.
1992 Czech And Slovak Olympiad IIIA, 2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$
1998 Taiwan National Olympiad, 6
In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.
2007 Today's Calculation Of Integral, 242
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
2019 MIG, 2
On Monday, Lyndon receives a $80$ on his daily math quiz. After being scolded by his parents, he works harder and gets an $83$ on Tuesday. From Tuesday onward, his score improves by $3$ points each day. What will be Lyndon's score that Friday?
$\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }95\qquad\textbf{(D) }98\qquad\textbf{(E) }100$
2001 National Olympiad First Round, 4
How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2022 Saudi Arabia JBMO TST, 4
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$
1989 IMO Longlists, 36
Connecting the vertices of a regular $ n$-gon we obtain a closed (not necessarily convex) $ n$-gon. Show that if $ n$ is even, then there are two parallel segments among the connecting segments and if $ n$ is odd then there cannot be exactly two parallel segments.
2020 Durer Math Competition Finals, 4
We have a positive integer $n$, whose sum of digits is $100$ . If the sum of digits of $44n$ is $800$ then what is the sum of digits of $3n$?
2015 Oral Moscow Geometry Olympiad, 5
A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.