Found problems: 85335
1993 Denmark MO - Mohr Contest, 3
Determine all real solutions $x,y$ to the system of equations
$$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$
2004 Thailand Mathematical Olympiad, 6
Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.
1997 Akdeniz University MO, 4
A plane dividing like a chessboard and write a real number each square such that, for a squares' number equal to its up, down ,left and right squares' numbers arithmetic mean. Prove that all number are equal.
2019 MIG, 7
How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
2022 BMT, 3
Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)
III Soros Olympiad 1996 - 97 (Russia), 9.6
Find the common fraction with the smallest positive denominator lying between the fractions $\frac{96}{35}$ and $\frac{97}{36} $.
2014 Austria Beginners' Competition, 3
Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$.
Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result.
(R. Henner, Vienna)
2010 CHMMC Winter, 10
Compute the number of $10$-bit sequences of $0$’s and $1$’s do not contain $001$ as a subsequence.
2005 National Olympiad First Round, 14
We call a number $10^3 < n < 10^6$ a [i]balanced [/i]number if the sum of its last three digits is equal to the sum of its other digits. What is the sum of all balanced numbers in $\bmod {13}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$
2012 AMC 10, 20
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2021 BMT, 20
For some positive integer $n$, $(1 + i) + (1 + i)^2 + (1 + i)^3 + ... + (1 + i)^n = (n^2 - 1)(1 - i)$, where $i = \sqrt{-1}$. Compute the value of $n$.
2011 IFYM, Sozopol, 1
$In$ $triangle$ $ABC$ $bisectors$ $AA_1$, $BB_1$ $and$ $CC_1$ $are$ $drawn$. $Bisectors$ $AA_1$ $and$ $CC_1$ $intersect$ $segments$ $C_1B_1$ $and$ $B_1A_1$ $at$ $points$ $M$ $and$ $N$, $respectively$. $Prove$ $that$ $\angle$$MBB_1$ = $\angle$$NBB_1$.
2014 Contests, 1
Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called[i] beautiful[/i] if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled.
[i]Proposed by Nairi M. Sedrakyan, Armenia[/i]
2023 Austrian MO Regional Competition, 2
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
2013 Romania National Olympiad, 3
Given $a\in (0,1)$ and $C$ the set of increasing functions
$f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine:
$(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$
$(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$
1956 AMC 12/AHSME, 25
The sum of all numbers of the form $ 2k \plus{} 1$, where $ k$ takes on integral values from $ 1$ to $ n$ is:
$ \textbf{(A)}\ n^2 \qquad\textbf{(B)}\ n(n \plus{} 1) \qquad\textbf{(C)}\ n(n \plus{} 2) \qquad\textbf{(D)}\ (n \plus{} 1)^2 \qquad\textbf{(E)}\ (n \plus{} 1)(n \plus{} 2)$
2007 Estonia National Olympiad, 5
Juhan wants to order by weight five balls of pairwise different weight, using only a balance scale. First, he labels the balls with numbers 1 to 5 and creates a list of weighings, such that each element in the list is a pair of two balls. Then, for every pair in the list, he weighs the two balls against each other. Can Juhan sort the balls by weight, using a list with less than 10 pairs?
2009 Sharygin Geometry Olympiad, 19
Given convex $ n$-gon $ A_1\ldots A_n$. Let $ P_i$ ($ i \equal{} 1,\ldots , n$) be such points on its boundary that $ A_iP_i$ bisects the area of polygon. All points $ P_i$ don't coincide with any vertex and lie on $ k$ sides of $ n$-gon. What is the maximal and the minimal value of $ k$ for each given $ n$?
2016 Sharygin Geometry Olympiad, P21
The areas of rectangles $P$ and $Q$ are equal, but the diagonal of $P$ is greater. Rectangle $Q$ can be covered by two copies of $P$. Prove that $P$ can be covered by two copies of $Q$.
1955 AMC 12/AHSME, 16
The value of $ \frac{3}{a\plus{}b}$ when $ a\equal{}4$ and $ b\equal{}\minus{}4$ is:
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \frac{3}{8} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \text{any finite number} \qquad
\textbf{(E)}\ \text{meaningless}$
1961 AMC 12/AHSME, 4
Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 . . .$. If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under:
${{ \textbf{(A)}\ \text{Addition}\qquad\textbf{(B)}\ \text{Multiplication} \qquad\textbf{(C)}\ \text{Division} \qquad\textbf{(D)}\ \text{Extraction of a positive integral root} }\qquad\textbf{(E)} \text{None of these} } $
2004 Switzerland - Final Round, 8
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.
2018 Ramnicean Hope, 2
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $
[i]Ovidiu Țâțan[/i]
2010 China Team Selection Test, 1
Let $\triangle ABC$ be an acute triangle, and let $D$ be the projection of $A$ on $BC$. Let $M,N$ be the midpoints of $AB$ and $AC$ respectively. Let $\Gamma_1$ and $\Gamma_2$ be the circumcircles of $\triangle BDM$ and $\triangle CDN$ respectively, and let $K$ be the other intersection point of $\Gamma_1$ and $\Gamma_2$. Let $P$ be an arbitrary point on $BC$ and $E,F$ are on $AC$ and $AB$ respectively such that $PEAF$ is a parallelogram. Prove that if $MN$ is a common tangent line of $\Gamma_1$ and $\Gamma_2$, then $K,E,A,F$ are concyclic.
1987 IMO Longlists, 4
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that
\[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\]
and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$