Found problems: 85335
2017 Princeton University Math Competition, B2
Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)
2004 Alexandru Myller, 2
The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular.
Show that the edges from $ A $ have equal lengths.
[i]Dinu Șerbănescu[/i]
2017 Stars of Mathematics, 4
Let $ ABC $ be an acute triangle having $ AB<AC, $ let $ M $ be the midpoint of the segment $ BC, D$ be a point on the segment $ AM, E $ be a point on the segment $ BD $ and $ F $ on the line $ AB $ such that $ EF $ is parallel to $ BC, $ and such that $ AE $ and $ DF $ pass through the orthocenter of $ ABC. $
Prove that the interior bisectors of $ \angle BAC $ and $ \angle BDC, $ together with $ BC $ are concurrent.
[i]Vlad Robu[/i]
2003 IMC, 4
Find all the positive integers $n$ for which there exists a Family $\mathcal{F}$ of three-element subsets of $S=\{1,2,...,n\}$ satisfying
\[\text{(i) for any two different elements $a,b \in S$ there exists exactly one $A \in \mathcal{F}$ containing both $a$ and $b$;}\]
\[\text{(ii) if $a,b,c,x,y,z$ are elements of $S$ such that $\{a,b,x\},\{a,c,y\},\{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F} $ }.\]
2014 Harvard-MIT Mathematics Tournament, 4
Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
2019 Estonia Team Selection Test, 12
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2007 National Olympiad First Round, 14
What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$?
$
\textbf{(A)}\ 49
\qquad\textbf{(B)}\ 53
\qquad\textbf{(C)}\ 97
\qquad\textbf{(D)}\ 103
\qquad\textbf{(E)}\ \text{None of the above}
$
2018 Saudi Arabia JBMO TST, 3
The cube $nxnxn$ consists of $n^3$
unit cubes $1x1x1$, and at least
one of these unit cubes is black. Show that we can always cut the cube in $2$ parallelepiped pieces so that each piece contains exactly one black 1x1 square .
2011 BAMO, 3
Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$.
Prove that the elements of $S$ may be arranged in an arithmetic progression.
This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.
2018 CMIMC Geometry, 5
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4. What is the magnitude of the $z$-coordinate of $T_0$?
2016 Finnish National High School Mathematics Comp, 4
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?
2011 SEEMOUS, Problem 2
Let $A=(a_{ij})$ be a real $n\times n$ matrix such that $A^n\ne0$ and $a_{ij}a_{ji}\le0$ for all $i,j$. Prove that there exist two nonreal numbers among eigenvalues of $A$.
2013 Switzerland - Final Round, 9
Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$
2002 Korea Junior Math Olympiad, 1
Find the value of $x^2+y^2+z^2$ where $x, y, z$ are non-zero and satisfy the following:
(1) $x+y+z=3$
(2) $x^2(\frac{1}{y}+\frac{1}{z})+y^2(\frac{1}{z}+\frac{1}{x})+z^2(\frac{1}{x}+\frac{1}{y})=-3$
2011 Romanian Masters In Mathematics, 3
A triangle $ABC$ is inscribed in a circle $\omega$.
A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$).
Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$.
Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$.
[i](Russia) Vasily Mokin and Fedor Ivlev[/i]
2013 India PRMO, 1
What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?
2003 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.
1972 Swedish Mathematical Competition, 4
Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$.
Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[
0.47 < \log_{10} 3 < 0.482.
\]
2018 IOM, 5
Ann and Max play a game on a $100 \times 100$ board.
First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once.
Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square.
Max wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies?
[i]Lev Shabanov[/i]
2002 Moldova National Olympiad, 3
Let $ a,b,c>0$. Prove that:
$ \dfrac{a}{2a\plus{}b}\plus{}\dfrac{b}{2b\plus{}c}\plus{}\dfrac{c}{2c\plus{}a}\leq 1$
2019 All-Russian Olympiad, 3
An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.
1990 All Soviet Union Mathematical Olympiad, 528
Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?
2019 Germany Team Selection Test, 2
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2019 Bundeswettbewerb Mathematik, 3
Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.
2017 Argentina National Math Olympiad Level 2, 6
In the governor elections, there were three candidates: $A$, $B$, and $C$. In the first round, $A$ received $44\%$ of the votes that were cast between $B$ and $C$. No candidate obtained the majority needed to win in the first round, and $C$ was the one who received the least votes of the three, so there was a runoff between $A$ and $B$. The voters for the runoff were the same as in the first round, except for $p\%$ of those who voted for $C$, who chose not to participate in the runoff; $p$ is an integer, $1 \leqslant p \leqslant 100$. It is also known that all those who voted for $B$ in the first round also voted for him again in the runoff, but it is unknown what those who voted for $A$ in the first round did.
A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of $p$ the journalist is telling the truth.
[b]Note:[/b] The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.