Found problems: 85335
1996 Baltic Way, 3
Let $ABCD$ be a unit square and let $P$ and $Q$ be points in the plane such that $Q$ is the circumcentre of triangle $BPC$ and $D$ be the circumcentre of triangle $PQA$. Find all possible values of the length of segment $PQ$.
2010 Laurențiu Panaitopol, Tulcea, 1
Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence
$$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$
tends to $ 2. $
[i]Mihai Bălună[/i]
2008 APMO, 5
Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$
be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.
2015 CCA Math Bonanza, T9
The Fibonacci numbers are defined as the sequence $F_n$ with $F_0=1$, $F_1=1$ and $F_{n+2} = F_{n+1} +F_n$. How many ways can $10$ be written as an ordered sum of numbers found in the Fibonacci sequence? For example, $3$ can be written as $1 + 1 + 1$, $2 + 1$, $1 + 2$, and $3$, for a total of $4$ ways.
[i]2015 CCA Math Bonanza Team Round #9[/i]
2013 AIME Problems, 1
The AIME Triathlon consists of a half-mile swim, a $30$-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
2010 Kazakhstan National Olympiad, 5
Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$.
Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$
2019 Taiwan APMO Preliminary Test, P6
Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$
(1)Find $f(0)$
(2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$
2011 Serbia National Math Olympiad, 3
Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.
Denmark (Mohr) - geometry, 1998.3
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
[img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]
2019 India PRMO, 13
Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i <
j \leq 101} x_i x_j$ ?
2010 Today's Calculation Of Integral, 653
If you're interested in participating in COP 4th, please sign up here by putting "I'm in" or "Yes!" or et cetera.
Thanks.
2014 ASDAN Math Tournament, 4
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
2022 Purple Comet Problems, 11
For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$
1989 All Soviet Union Mathematical Olympiad, 510
A convex polygon is such that any segment dividing the polygon into two parts of equal area which has at least one end at a vertex has length $< 1$. Show that the area of the polygon is $< \pi /4$.
2006 International Zhautykov Olympiad, 2
Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality:
\[ (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd).
\]
2004 JBMO Shortlist, 4
Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.
1999 Moldova Team Selection Test, 1
Let $a, b, c, d, e$ $(a < b < c < d < e)$be positive integers. FInd the greatest possible value of the expression $\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]}$, where $[x,y]$ denotes the least common multiple of $x{}$ and $y{}$.
2007 Indonesia TST, 1
Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.
2008 Harvard-MIT Mathematics Tournament, 32
Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.
2010 Today's Calculation Of Integral, 665
Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$.
[i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]
1996 Chile National Olympiad, 6
Two circles, $C$ and $K$, are secant at $A$ and $B$. Let $P$ be a point on the arc $AB$ of $C$. Lines $PA$ and $PB$ intersect $K$ again at $R$ and $S$ respectively. Let $P'$ be another point at same arc as $P$, so that lines $P'A$ and $P'B$ again intersect $K$ at $R'$ and $S'$, respectively. Prove that the arcs $RS$ and $R'S'$ have equal measures.
[img]https://cdn.artofproblemsolving.com/attachments/2/4/88693c36159179fb2b098b671a2f8281b37aae.png[/img]
2005 MOP Homework, 7
Let $x_{1,1}$, $x_{2,1}$, ..., $x_{n,1}$, $n \ge 2$, be a sequence of integers and assume that not all $x_{i,1}$ are equal. For $k \ge 2$, if sequence $\{x_{i,k}\}^n_{i=1}$ is defined, we define sequence $\{x_{i,k+1}\}^n_{i=1}$ as \[x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}),\] for $i=1, 2, ..., n$, (where $x_{n+1,k}=x_{1,k}$). Show that if $n$ is odd then there exist indices $j$ and $k$ such that $x_{j,k}$ is not an integer.
2024 Indonesia TST, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
2003 Estonia Team Selection Test, 3
Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows:
(i) $A(0, m, n) = m'$ for all $m, n \in N$
(ii) $A(k', 0, n) =\left\{ \begin{array}{ll}
n & if \, \, k = 0 \\
0 & if \, \,k = 1, \\
1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$
(iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$.
Compute $A(5, 3, 2)$.
(H. Nestra)
2019 Bulgaria National Olympiad, 6
Let $ABCDEF$ be an inscribed hexagon with
$$AB.CD.EF=BC.DE.FA$$
Let $B_1$ be the reflection point of $B$ with respect to $AC$ and $D_1$ be the reflection point of $D$ with respect to $CE,$ and finally let $F_1$ be the reflection point of $F$ with respect to $AE.$ Prove that $\triangle B_1D_1F_1\sim BDF.$