Found problems: 85335
2001 Grosman Memorial Mathematical Olympiad, 1
Find all real solutions of the system
$$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$
1991 Tournament Of Towns, (305) 2
In $\vartriangle ABC$, $AB = AC$ and $\angle BAC = 20^o$. A point $D$ lies on the side $AB$ and $AD = BC$. Find $\angle BCD$.
(LF. Sharygin, Moscow)
2017 Princeton University Math Competition, B1
If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
1966 Polish MO Finals, 1
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$
2003 Abels Math Contest (Norwegian MO), 2a
Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$
2009 Belarus Team Selection Test, 1
Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.
I. Voronovich
2015 Iran Geometry Olympiad, 2
Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.
by Mahdi Etesami Fard
2004 ITAMO, 2
Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.
VMEO I 2004, 6
Consider all binary sequences of length $n$. In a sequence that allows the interchange of positions of an arbitrary set of $k$ adjacent numbers, ($k < n$), two sequences are said to be [i]equivalent [/i] if they can be transformed from one sequence to another by a finite number of transitions as above. Find the number of sequences that are not equivalent.
2023 BMT, 12
Call an $n$-digit integer with distinct digits [i]mountainous [/i]if, for some integer $1 \le k \le n$, the first $k$ digits are in strictly ascending order and the following $n - k$ digits are in strictly descending order. How many $5$-digit mountainous integers with distinct digits are there?
2016 BMT Spring, 2
Define $a \star b$ to be $2ab + a + b$. What is $((3 \star 4) \star 5) - (4 \star (5 \star 3))$ ?
1956 Moscow Mathematical Olympiad, 335
a) $100$ numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive.
Can the sum of all underlined numbers be
(i) negative?
(ii) equal to zero?
b) $n$ numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.
1992 Poland - First Round, 5
Given is a halfplane with points $A$ and $C$ on its edge. For every point $B$ on this halfplane consider the squares $ABKL$ and $BCMN$ lying outside the triangle $ABC$. Prove that all the lines $LM$ (as the point $B$ varies) have a common point.
1958 Miklós Schweitzer, 10
[b]10.[/b] Prove that the function
$f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta$
where $k$ is a positive integer, satisfies the following conditions:
[b](i)[/b] $f(x)=0$ if $\mid x \mid \geq k$ and $f(x) \geq 0$ elsewhere;
[b](ii)[/b] in interval $(l,l+1)$ $(l= -k, -k+1, \dots , k-1)$ the function $f(x)$ is a polynomial of degree $2k-1$ at most. [b](R. 7)[/b]
2013 AMC 8, 19
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, ``I didn't get the lowest score in our class,'' and Bridget adds, ``I didn't get the highest score.'' What is the ranking of the three girls from highest to lowest?
$\textbf{(A)}\ \text{Hannah, Cassie, Bridget} \qquad \textbf{(B)}\ \text{Hannah, Bridget, Cassie}$ \\ $\qquad \textbf{(C)}\ \text{Cassie, Bridget, Hannah} \qquad \textbf{(D)}\ \text{Cassie, Hannah, Bridget}$ \\$ \qquad \textbf{(E)}\ \text{Bridget, Cassie, Hannah}$
2024 All-Russian Olympiad, 8
$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.)
[i]Proposed by I. Bogdanov[/i]
2006 Finnish National High School Mathematics Competition, 5
The game of Nelipe is played on a $16\times16$-grid as follows: The two players write in turn numbers $1, 2,..., 16$ in diff erent squares. The numbers on each row, column, and in every one of the 16 smaller squares have to be di fferent. The loser is the one who is not able to write a number. Which one of the players wins, if both play with an optimal strategy?
1971 IMO Longlists, 30
Prove that the system of equations
\[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \]
$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
2008 ITest, 84
Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.
2020 BMT Fall, 8
Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$.
2012 Tournament of Towns, 5
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.
2024 HMNT, 13
Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$
2018 AMC 10, 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2010 Harvard-MIT Mathematics Tournament, 3
Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\ldots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}<i$ and $-1$ if $S_{i-1}\geq i$. What is the largest $k\leq 2010$ such that $S_k=0$?
2018 Stanford Mathematics Tournament, 7
Two equilateral triangles $ABC$ and $DEF$, each with side length $1$, are drawn in $2$ parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon $AF BDCE$. Line segments $AE$, $AF$, $BF$, $BD$, $CD,$ and $CE$ are drawn, and suppose that each of these segments also has length $1$. Compute the volume of the resulting solid that is formed.