Found problems: 85335
2012 Online Math Open Problems, 9
Define a sequence of integers by $T_1 = 2$ and for $n\ge2$, $T_n = 2^{T_{n-1}}$. Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
[i]Ray Li.[/i]
2005 Argentina National Olympiad, 1
Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.
1999 All-Russian Olympiad Regional Round, 11.5
Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds:
$$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$
2005 National Olympiad First Round, 26
For every positive integer $n$, $f(2n+1)=2f(2n)$, $f(2n)=f(2n-1)+1$, and $f(1)=0$. What is the remainder when $f(2005)$ is divided by $5$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
2004 Kazakhstan National Olympiad, 8
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.
2013 Princeton University Math Competition, 2
What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?
2024 Iran MO (3rd Round), 2
For all positive integers $n$ Prove that one can find pairwise coprime integers $a,b,c>n$ such that the set of prime divisors of the numbers $a+b+c$ and $ab+bc+ac$ coincides.
Proposed by [i]Mohsen Jamali[/i] and [i]Hesam Rajabzadeh[/i]
1996 Tournament Of Towns, (489) 2
An exterior common tangent to two non-intersecting circles with centers and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point on the line $A_1A_2$ such that $CD$ is perpendicular to $B_1B_2$. Prove that $A_1D = DA_2$.
2009 China Girls Math Olympiad, 1
Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$
2014 Macedonia National Olympiad, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
1963 Polish MO Finals, 2
In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.
2017 Estonia Team Selection Test, 2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
LMT Team Rounds 2010-20, A6 B17
Circle $\omega$ has radius 10 with center $O$. Let $P$ be a point such that $PO=6$. Let the midpoints of all chords of $\omega$ through $P$ bound a region of area $R$. Find the value of $\lfloor 10R \rfloor$.
[i]Proposed by Andrew Zhao[/i]
2000 Harvard-MIT Mathematics Tournament, 34
What is the largest $n$ such that $n! + 1$ is a square?
2016 IOM, 2
Let $a_1, . . . , a_n$ be positive integers satisfying the inequality
$\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$.
Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
2009 Ukraine National Mathematical Olympiad, 4
In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$
1974 All Soviet Union Mathematical Olympiad, 194
Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.
1983 All Soviet Union Mathematical Olympiad, 366
Given a point $O$ inside triangle $ABC$ . Prove that $$S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$$
where $S_A, S_B, S_C$ denote areas of triangles $BOC, COA, AOB$ respectively.
2021 Saint Petersburg Mathematical Olympiad, 6
Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent.
[i]A. Kuznetsov[/i]
2013 NIMO Problems, 5
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?
[i]Based on a proposal by Ahaan Rungta[/i]
1960 Putnam, B1
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
2008 AMC 10, 18
Bricklayer Brenda would take $ 9$ hours to build a chimney alone, and bricklayer Brandon would take $ 10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $ 10$ bricks per hour. Working together, they build the chimney in $ 5$ hours. How many bricks are in the chimney?
$ \textbf{(A)}\ 500 \qquad
\textbf{(B)}\ 900 \qquad
\textbf{(C)}\ 950 \qquad
\textbf{(D)}\ 1000 \qquad
\textbf{(E)}\ 1900$
2021 China National Olympiad, 2
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
2011 Romania Team Selection Test, 2
In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.
2008 JBMO Shortlist, 1
Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.