Found problems: 85335
1949 Moscow Mathematical Olympiad, 157
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.
b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
1958 November Putnam, B7
Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.
2014 Contests, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
2023 Kazakhstan National Olympiad, 2
$a,b,c$ are positive real numbers such that $a+b+c\ge 3$ and $a^2+b^2+c^2=2abc+1$. Prove that $$a+b+c\le 2\sqrt{abc}+1$$
1963 AMC 12/AHSME, 7
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad
\textbf{(2)}\ -2x-3y=10 \qquad
\textbf{(3)}\ 3y+2x=12 \qquad
\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:
$\textbf{(A)}\ \text{(1) and (4)} \qquad
\textbf{(B)}\ \text{(1) and (3)} \qquad
\textbf{(C)}\ \text{(1) and (2)} \qquad
\textbf{(D)}\ \text{(2) and (4)} \qquad
\textbf{(E)}\ \text{(2) and (3)}$
2018 PUMaC Algebra B, 7
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
1982 National High School Mathematics League, 12
Given a circle $C:x^2+y^2=r^2$ ($r$ is an odd number). $P(u,v)\in C$, satisfying: $u=p^m, v=q^n$($p,q$ are prime numbers, $m,n$ are integers, $u>v$).
Define $A,B,C,D,M,N:A(r,0),B(-r,0),C(0,-r),D(0,r),M(u,0),N(0,v)$.
Prove that $|AM|=1,|BM|=9,|CN|=8,|DN|=2$.
1977 Czech and Slovak Olympiad III A, 6
A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.
2018 Baltic Way, 7
On a $16 \times 16$ torus as shown all $512$ edges are colored red or blue. A coloring is [i]good [/i]if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the $4$ edges of an arbitrary cell. What is the largest number of good colorings so that none of them can be converted to another by a sequence of moves?
2000 Pan African, 1
Solve for $x \in R$:
\[ \sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x} \]
2010 Rioplatense Mathematical Olympiad, Level 3, 1
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.
1957 AMC 12/AHSME, 19
The base of the decimal number system is ten, meaning, for example, that $ 123 \equal{} 1\cdot 10^2 \plus{} 2\cdot 10 \plus{} 3$. In the binary system, which has base two, the first five positive integers are $ 1,\,10,\,11,\,100,\,101$. The numeral $ 10011$ in the binary system would then be written in the decimal system as:
$ \textbf{(A)}\ 19 \qquad
\textbf{(B)}\ 40\qquad
\textbf{(C)}\ 10011\qquad
\textbf{(D)}\ 11\qquad
\textbf{(E)}\ 7$
2008 Romania Team Selection Test, 1
Let $ ABC$ be a triangle with $ \measuredangle{BAC} < \measuredangle{ACB}$. Let $ D$, $ E$ be points on the sides $ AC$ and $ AB$, such that the angles $ ACB$ and $ BED$ are congruent. If $ F$ lies in the interior of the quadrilateral $ BCDE$ such that the circumcircle of triangle $ BCF$ is tangent to the circumcircle of $ DEF$ and the circumcircle of $ BEF$ is tangent to the circumcircle of $ CDF$, prove that the points $ A$, $ C$, $ E$, $ F$ are concyclic.
[i]Author: Cosmin Pohoata[/i]
2003 Croatia National Olympiad, Problem 3
The natural numbers $1$ through $2003$ are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is $k$, the order of the first $k$ terms is reversed. Prove that after several operations number $1$ will occur on the first place.
2008 Iran Team Selection Test, 10
In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI\equal{}90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B\equal{}\angle IPB$, find the angle $ \angle A$.
2008 AMC 12/AHSME, 16
The numbers $ \log(a^3b^7)$, $ \log(a^5b^{12})$, and $ \log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $ 12^\text{th}$ term of the sequence is $ \log{b^n}$. What is $ n$?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 76 \qquad
\textbf{(D)}\ 112 \qquad
\textbf{(E)}\ 143$
1997 Baltic Way, 10
Prove that in every sequence of $79$ consecutive positive integers written in the decimal system, there is a positive integer whose sum of digits is divisible by $13$.
2000 Slovenia National Olympiad, Problem 3
A point $D$ is taken inside an isosceles triangle $ABC$ with base $AB$ and $\angle C=80^\circ$ such that $\angle DAB=10^\circ$ and $\angle DBA=20^\circ$. Compute $\angle ACD$.
2014 Purple Comet Problems, 19
Let $n$ be a positive integer such that $\lfloor\sqrt n\rfloor-2$ divides $n-4$ and $\lfloor\sqrt n\rfloor+2$ divides $n+4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x\rfloor$ refers to the greatest integer less than or equal to $x$.)
PEN R Problems, 3
Prove no three lattice points in the plane form an equilateral triangle.
2006 Moldova Team Selection Test, 3
Let $a,b,c$ be sides of the triangle. Prove that
\[ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . \]
2016 Swedish Mathematical Competition, 1
In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this?
[img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]
2010 Contests, 1
$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$
1997 Poland - Second Round, 4
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.
1969 AMC 12/AHSME, 28
Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of the squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }4\qquad
\textbf{(E) }\text{infinite}$