Found problems: 85335
Ukrainian TYM Qualifying - geometry, 2013.17
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
Kvant 2024, M2825
At the same time, three beetles with identical speeds began to crawl along the heights of an acute-angled non-isosceles triangle from its vertices. At some point, it turned out that the first and second beetles were on a circle inscribed in a triangle. Prove that at this moment the third beetle is also on this circle.
[i]A. Kuznetsov[/i]
2004 China Second Round Olympiad, 3
For integer $n\ge 4$, find the minimal integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.
2009 Austria Beginners' Competition, 3
There are any number of stamps with the values $134$, $135$, $...$, $142$ and $143$ cents available. Find the largest integer value (in cents) that cannot be represented by these stamps.
(G. Woeginger, TU Eindhoven, The Netherlands)
[hide=original wording]Es stehen beliebig viele Briefmarken mit den Werten 134, 135. . .., 142 und 143 Cent zur Verfügung. Man bestimme den größten ganzzahligen Wert (in Cent), der nicht durch diese Briefmarken dargestellt werden kann.[/hide]
2008 Harvard-MIT Mathematics Tournament, 10
Let $ ABC$ be a triangle with $ BC \equal{} 2007$, $ CA \equal{} 2008$, $ AB \equal{} 2009$. Let $ \omega$ be an excircle of $ ABC$ that touches the line segment $ BC$ at $ D$, and touches extensions of lines $ AC$ and $ AB$ at $ E$ and $ F$, respectively (so that $ C$ lies on segment $ AE$ and $ B$ lies on segment $ AF$). Let $ O$ be the center of $ \omega$. Let $ \ell$ be the line through $ O$ perpendicular to $ AD$. Let $ \ell$ meet line $ EF$ at $ G$. Compute the length $ DG$.
2004 National Olympiad First Round, 24
What is the sum of cubes of real roots of the equation $x^3-2x^2-x+1=0$?
$
\textbf{(A)}\ -6
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ \text{None of above}
$
2023 UMD Math Competition Part I, #15
What is the least positive integer $m$ such that the following is true?
[i]Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$ [/i] \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~1415\]
2021 Science ON all problems, 4
$\textbf{(a)}$ On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite.
$\textbf{(b)}$ Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
[i] (Călin Udrea) [/i]
1972 Poland - Second Round, 4
A cube with edge length $ n $ is divided into $ n^3 $ unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?
1976 Miklós Schweitzer, 3
Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that
a) $ n! \in H$ for all sufficiently large $ n$,
b)$ H$ has density $ 0$.
[i]P. Erdos[/i]
1987 AMC 8, 4
Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?
$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$
2016 Harvard-MIT Mathematics Tournament, 15
Compute $\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)$.
1985 Traian Lălescu, 1.3
Let $ G $ be a finite group of odd order having, at least, three elements. For $ a\in G $ denote $ n(a) $ as the number of ways $ a $ can be written as a product of two distinct elements of $ G. $
Prove that $ \sum_{\substack{a\in G\\a\neq\text{id}}} n(a) $ is a perfect square.
2004 Iran Team Selection Test, 2
Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.
2018 Pan-African Shortlist, G6
Let $\Gamma$ be the circumcircle of an acute triangle $ABC$. The perpendicular line to $AB$ passing through $C$ cuts $AB$ in $D$ and $\Gamma$ again in $E$. The bisector of the angle $C$ cuts $AB$ in $F$ and $\Gamma$ again in $G$. The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I$. Prove that $AI = EB$.
2008 AMC 10, 12
In a collection of red, blue, and green marbles, there are $ 25\%$ more red marbles than blue marbles, and there are $ 60\%$ more green marbles than red marbles. Suppose that there are $ r$ red marbles. What is the total number of marbles in that collection?
$ \textbf{(A)}\ 2.85r \qquad \textbf{(B)}\ 3r \qquad \textbf{(C)}\ 3.4r \qquad \textbf{(D)}\ 3.85r \qquad \textbf{(E)}\ 4.25r$
2021 Czech-Polish-Slovak Junior Match, 5
Find all three real numbers $(x, y, z)$ satisfying the system of equations $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}$$ $$x^2 + y^2 + z^2 = xy + yz + zx + 4$$
2019 Math Prize for Girls Problems, 10
A $1 \times 5$ rectangle is split into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?
PEN D Problems, 5
Prove that for $n\geq 2$, \[\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.\]
1910 Eotvos Mathematical Competition, 1
If $a, b, c$ are real numbers such that $$a^2 + b^2 + c^2 = 1$$ prove the inequalities $$- \frac12 \le ab + bc + ca \le 1$$
2025 Bangladesh Mathematical Olympiad, P5
Mugdho and Dipto play a game on a numbered row of $n \geq 5$ squares. At the beginning, a pebble is put on the first square and then the players make consecutive moves; Mugdho starts. During a move a player is allowed to choose one of the following:
[list]
[*] move the pebble one square rightward
[*] move the pebble four squares rightward
[*] move the pebble two squares leftward
[/list]
All of the possible moves are only allowed if the pebble stays within the borders of the square row. The player who moves the pebble to the last square (a. k. a $n$-th) wins. Determine for which values of $n$ each of the players has a winning strategy.
1973 Swedish Mathematical Competition, 1
$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).
2004 China Team Selection Test, 2
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.
PEN N Problems, 11
The infinite sequence of 2's and 3's \[\begin{array}{l}2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ 3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\cdots \end{array}\] has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1+\lfloor rm \rfloor$ for some nonnegative integer $m$.
2020 Kyiv Mathematical Festival, 1.1
(a) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = \frac12 a_{k- 1} + \frac12 a_{k+1 }$$
(b) Find the numbers $a_0,. . . , a_{100}$, such that $a_0 = 0, a_{100} = 1$ and for all $k = 1,. . . , 99$ :
$$a_k = 1+\frac12 a_{k- 1} + \frac12 a_{k+1 }$$.