Found problems: 85335
1982 Tournament Of Towns, (027) 1
Prove that for all natural numbers $n$ greater than $1$ :
$$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$
(VV Kisil)
2019 Denmark MO - Mohr Contest, 1
Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?
Oliforum Contest I 2008, 1
(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
(b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.
2006 AIME Problems, 13
For each even positive integer $x$, let $g(x)$ denote the greatest power of $2$ that divides $x$. For example, $g(20)=4$ and $g(16)=16$. For each positive integer $n$, let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than $1000$ such that $S_n$ is a perfect square.
1953 Poland - Second Round, 3
A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.
2023 Tuymaada Olympiad, 8
Circle $\omega$ lies inside the circle $\Omega$ and touches it internally at point $P$. Point $S$ is taken on $\omega$ and the tangent to $\omega$ is drawn through it. This tangent meets $\Omega$ at points $A$ and $B$. Let $I$ be the centre of $\omega$. Find the locus of circumcentres of triangles $AIB$.
2014 Harvard-MIT Mathematics Tournament, 19
Let $ABCD$ be a trapezoid with $AB\parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and $EDCF$ have areas $24$ and $36$, respectively, and triangle $ABH$ has area $25$. Find the area of triangle $CDG$.
2010 HMNT, 8
Allison has a coin which comes up heads $\frac23$ of the time. She flips it $5$ times. What is the probability that she sees more heads than tails?
1952 Moscow Mathematical Olympiad, 227
$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.
2009 All-Russian Olympiad, 7
We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?
LMT Team Rounds 2021+, 3
Let the four real solutions to the equation $x^2 + \frac{144}{x^2} = 25$ be $r_1, r_2, r_3$, and $r_4$. Find $|r_1| +|r_2| +|r_3| +|r_4|$.
1997 AMC 8, 17
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("$x$",(2.75,3.5),NNE);
label("$y$",(4.125,1.5),NNE);
[/asy]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$
1999 Hong kong National Olympiad, 4
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+yf(x))=f(x)+xf(y) \quad \text{for all}\ x,y \in\mathbb{R}\]
1995 Taiwan National Olympiad, 3
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.
1971 IMO Longlists, 14
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
2005 Polish MO Finals, 3
Let be a convex polygon with $n >5$ vertices and area $1$. Prove that there exists a convex hexagon inside the given polygon with area at least $\dfrac{3}{4}$
2011 Morocco National Olympiad, 1
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
2016 Romanian Masters in Mathematic, 2
Given positive integers $m$ and $n \ge m$, determine the largest number of dominoes ($1\times2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2n$ columns consisting of cells ($1 \times 1$
squares) so that:
(i) each domino covers exactly two adjacent cells of the board;
(ii) no two dominoes overlap;
(iii) no two form a $2 \times 2$ square; and
(iv) the bottom row of the board is completely covered by $n$ dominoes.
2009 Today's Calculation Of Integral, 449
Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$
2020 Princeton University Math Competition, 3
Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form $2^a3^b$, where a and b are positive integers between $ 1$ and $2020$, inclusive. Each turn, Alice guess a number m, and Bob will tell her either $\gcd (m, n)$ or $lcm (m, n)$ (letting her know that he is saying that $gcd$ or $lcm$), as well as whether any of the respective powers match up in their prime factorization. In particular, if $m = n$, Bob will let Alice know this, and the game is over. Determine the smallest number $k$ so that Alice is always able to find $n$ within $k$ guesses, regardless of Bob’s number or choice of revealing either the $lcm$, or the $gcd$ .
2012 Traian Lălescu, 1
Let $a,b,c,\alpha,\beta,\gamma \in\mathbb{R}$ such as $a^2+b^2+c^2 \neq 0 \neq \alpha\beta\gamma$ and $24^{\alpha}\neq 3^{\beta} \neq 2012^{\gamma} \neq 24^{\alpha}$. Prove that the equation \[ a \cdot 24^{\alpha x}+b \cdot 3^{\beta x} + c \cdot 2012^{\gamma x}=0 \] has at most two real solutions.
2003 Gheorghe Vranceanu, 4
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
2010 Mexico National Olympiad, 3
Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if
\[pqr\mid (pq)^r+(qr)^p+(rp)^q-1,\]
then
\[(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).\]
2004 Tournament Of Towns, 2
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?
2010 National Olympiad First Round, 33
Let $D$ be the midpoint of $[AC]$ of $\triangle ABC$ with $m(\widehat{ABC})=90^\circ$ and $|AC|=10$. Let $E$ be the point of intersections of bisectors of $[AD]$ and $[BD]$. Let $F$ be the point of intersections of bisectors of $[BD]$ and $[CD]$. If $|EF|=13$, then $|AB|$ can be
$ \textbf{(A)}\ 20\sqrt{\frac 2{13}}
\qquad\textbf{(B)}\ 15\sqrt{\frac 2{13}}
\qquad\textbf{(C)}\ 10\sqrt{\frac 2{13}}
\qquad\textbf{(D)}\ 5\sqrt{\frac 2{13}}
\qquad\textbf{(E)}\ \text{None}
$