Found problems: 85335
2015 Romanian Master of Mathematics, 6
Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that
$\bullet$ the sides of $T$ are parallel to the sides of $U$;
$\bullet$ the interior of $T$ contains exactly one point of $C$;
$\bullet$ the area of $T$ is at least $\mu$.
2020 IMC, 8
Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$
2000 Estonia National Olympiad, 5
Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician.
Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.
2021 Brazil National Olympiad, 1
Let \(ABCD\) be a convex quadrilateral in the plane and let \(O_{A}, O_{B}, O_{C}\) and \(O_{D}\) be the circumcenters of the triangles \(BCD, CDA, DAB\) and \(ABC\), respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.
2004 AMC 8, 17
Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen?
$\textbf{(A)}\ 1\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 12$
2008 Romania Team Selection Test, 1
Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.
2012 Princeton University Math Competition, A7 / B8
A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either:
$\bullet$ grades another problem of the same student, or
$\bullet$ grades the same problem of the student $s_{i-1}$ or $s_{i+1}$ (if $i > 1$ and $i < 40$, respectively).
He grades each problem exactly once, starting with the first problem of $s_1$ and ending with the third problem of $s_{40}$. Let $N$ be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when $N$ is divided by $100$.
2015 Bangladesh Mathematical Olympiad, 7
In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.
2017 China Team Selection Test, 2
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$
Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
2022-2023 OMMC, 25
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken?
$\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$
2009 USAMO, 5
Trapezoid $ ABCD$, with $ \overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$. Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively. Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively. Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.
2008 Danube Mathematical Competition, 2
In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.
2005 AIME Problems, 5
Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.
1957 AMC 12/AHSME, 47
In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);
real m = 0;
real b = 0;
pair O = origin;
pair X = (-1,0);
pair Y = (1,0);
pair Q = midpoint(O--X);
pair A = (Q.x, -1*sqrt(3)/2);
pair B = (Q.x, -1*A.y);
pair M = (Q.x + sqrt(3)/2,0);
m = (B.y - M.y)/(B.x - M.x);
b = (B.y - m*B.x);
pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
m = (A.y - M.y)/(A.x - M.x);
b = (A.y - m*A.x);
pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));
draw(Circle(O,1));
draw(Arc(Q,sqrt(3)/2,-90,90));
draw(A--B);
draw(X--Y);
draw(B--D);
draw(A--C);
draw(A--D);
dot(O);dot(M);
label("$B$",B,NW);
label("$C$",C,NE);
label("$Y$",Y,E);
label("$D$",D,SE);
label("$A$",A,SW);
label("$X$",X,W);
label("$Q$",Q,SW);
label("$O$",O,SW);
label("$M$",M,NE+2N);[/asy]$ \textbf{(A)}\ r\sqrt {2} \qquad \textbf{(B)}\ r\qquad \textbf{(C)}\ \text{not a side of an inscribed regular polygon}\qquad \textbf{(D)}\ \frac {r\sqrt {3}}{2}\qquad \textbf{(E)}\ r\sqrt {3}$
2019 Romania National Olympiad, 4
Let $A$ and $B$ be two nonempty finite sets of nonnegative integers. We denote by $\mathcal{F}$ the set of all functions $f:\mathcal{P}(A) \to B$ that satisfy
[center]$f(X\cap Y)=\min \{f(X), f(Y)\},$ for all $X,Y \subset A,$[/center]
and by $\mathcal{G}$ the set of all functions $g:\mathcal{P}(A) \to B$ that satisfy
[center]$g(X\cup Y)=\max \{g(X), g(Y)\},$ for all $X,Y \subset A.$[/center]
Prove that $\mathcal F$ and $\mathcal G$ have the same number of elements and find this number.
2000 Korea Junior Math Olympiad, 3
Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.
1998 Romania Team Selection Test, 3
Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials.
[i]Marius Cavachi[/i]
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2014 Greece JBMO TST, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
2025 Caucasus Mathematical Olympiad, 2
There are $30$ children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
2003 National Olympiad First Round, 35
$n+m-1$ unit squares are arranged in $L$-shape whose one side contains $n$ squares and the other side contains $m$ squares. Ayse and Betul plays a turn based game with following rules: Ayse plays first. At each move, the player captures desired number of adjacent squares in same side of $L$. The one who captures the last square loses the game. If four games are played for pairs $(n,m)=(2003,2003)$, $(2002,2003)$, $(2003,3)$, $(2001,2003)$; how many times can Ayse guarantee to win?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
1996 All-Russian Olympiad Regional Round, 8.5
Is it possible to arrange the chips in the cells of an $8 \times 8$ board so that in any two columns the number of chips is the same, and in any two lines are different?
2018 Kyiv Mathematical Festival, 1
A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?
2012 Federal Competition For Advanced Students, Part 1, 4
Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$.
Show that $\gamma$ is the second-largest angle in the triangle $ABC$.
2025 Malaysian APMO Camp Selection Test, 1
A sequence is defined as $a_1=2025$ and for all $n\ge 2$, $$a_n=\frac{a_{n-1}+1}{n}$$ Determine the smallest $k$ such that $\displaystyle a_k<\frac{1}{2025}$.
[i]Proposed by Ivan Chan Kai Chin[/i]