Found problems: 85335
2006 District Olympiad, 4
a) Prove that we can assign one of the numbers $1$ or $-1$ to the vertices of a cube such that the product of the numbers assigned to the vertices of any face is equal to $-1$.
b) Prove that for a hexagonal prism such a mapping is not possible.
2021 USMCA, 16
Let $\mathcal{C}$ be a right circular cone with height $\sqrt{15}$ and base radius $1$. Let $V$ be the vertex of $\mathcal{C}$, $B$ be a point on the circumference of the base of $\mathcal{C}$, and $A$ be the midpoint of $VB$. An ant travels at constant velocity on the surface of the cone from $A$ to $B$ and makes two complete revolutions around $\mathcal{C}$. Find the distance the ant travelled.
2006 JBMO ShortLists, 2
Let $ x,y,z$ be positive real numbers such that $ x\plus{}2y\plus{}3z\equal{}\frac{11}{12}$. Prove the inequality $ 6(3xy\plus{}4xz\plus{}2yz)\plus{}6x\plus{}3y\plus{}4z\plus{}72xyz\le \frac{107}{18}$.
1985 AIME Problems, 8
The sum of the following seven numbers is exactly 19:
\[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\]
\[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\]
It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?
2010 Tournament Of Towns, 2
Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$
2023 Federal Competition For Advanced Students, P2, 5
Let $ABC$ be an acute triangle with $AC\neq BC$, $M$ the midpoint of side $AB$, $H$ is the orthocenter of $\triangle ABC$, $D$ on $BC$ is the foot of the altitude from $A$ and $E$ on $AC$ is the foot of the perpendicular from $B$. Prove that the lines $AB, DE$ and the perpendicular to $MH$ through $C$ are concurrent.
Denmark (Mohr) - geometry, 2013.5
The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$.
[img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]
Mathley 2014-15, 3
In a triangle $ABC$, $D$ is the reflection of $A$ about the sideline $BC$. A circle $(K)$ with diameter $AD$ meets $DB,DC$ at $M,N$ which are distinct from $D$. Let $E,F$ be the midpoint of $CA,AB$. The circumcircles of $KEM,KFN$ meet each other again at $L$, distinct from $K$. Let $KL$ meets $EF$ at $X$; points $Y,Z$ are defined in the same manner. Prove that three lines $AX,BY,CZ$ are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
V Soros Olympiad 1998 - 99 (Russia), 9.4
There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than $100$ different ones. What is the largest possible value for $n$?
1995 Canada National Olympiad, 2
Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.
2016 Hanoi Open Mathematics Competitions, 7
Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?
2012 Sharygin Geometry Olympiad, 15
Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.
2021 All-Russian Olympiad, 6
Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$, and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$. Let $M$ the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.
2021 Peru PAGMO TST, P7
In a country there are $2021$ cities. Each pair of cities is either linked by a single road or not linked at all. It is known that for any subset of $2019$ cities, the total number of roads between them is the same. If the total number of roads in that country is $A$, find all possible values of $A$.
Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31
The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.
1969 AMC 12/AHSME, 9
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:
$\textbf{(A) }27\qquad
\textbf{(B) }27\tfrac14\qquad
\textbf{(C) }27\tfrac12\qquad
\textbf{(D) }28\qquad
\textbf{(E) }28\tfrac12$
2014 ELMO Shortlist, 9
Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]
2012 Dutch IMO TST, 3
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.
1978 Romania Team Selection Test, 1
Prove that for every partition of $ \{ 1,2,3,4,5,6,7,8,9\} $ into two subsets, one of the subsets contains three numbers such that the sum of two of them is equal to the double of the third.
2003 JHMMC 8, 30
Calculate $1 + 3 + 5 +\cdots+ 195 + 197 + 199$
2024 Iran MO (3rd Round), 6
Sequence of positive integers $\{x_k\}_{k\geq 1}$ is given such that $x_1=1$ and for all $n\geq 1$ we have
$$x_{n+1}^2+P(n)=x_n x_{n+2}$$
where $P(x)$ is a polynomial with non-negative integer coefficients. Prove that $P(x)$ is the constant polynomial.
Proposed by [i]Navid Safaei[/i]
1991 Baltic Way, 6
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)
2005 AMC 10, 1
A scout troop buys $ 1000$ candy bars at a price of five for $ \$2$. They sell all the candy bars at a price of two for $ \$1$. What was their profit, in dollars?
$ \textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 200 \qquad
\textbf{(C)}\ 300 \qquad
\textbf{(D)}\ 400 \qquad
\textbf{(E)}\ 500$
1997 Chile National Olympiad, 1
Lautaro, Camilo and Rafael give the same exams. Each note is a positive integer. Camilo was the first in physics. Lautaro obtained a total score of $20$, Camilo, a total of $10$ and Rafael, a total of $9$. Among all the tests, there were no two scores that were repeated. Determine how many They took exams, and who was second in math.
2007 India National Olympiad, 4
Let $ \sigma = (a_1, a_2, \cdots , a_n)$ be permutation of $ (1, 2 ,\cdots, n)$. A pair $ (a_i, a_j)$ is said to correspond to an [b]inversion[/b] of $\sigma$ if $ i<j$ but $ a_i>a_j$. How many permutations of $ (1,2,\cdots,n)$, $ n \ge 3$, have exactly [b]two[/b] inversions?
For example, In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $ (2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.