Found problems: 85335
2020 Simon Marais Mathematics Competition, B3
A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\]
At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$;
$c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists.
At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$.
Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.
2023 Macedonian Mathematical Olympiad, Problem 5
There are $n$ boys and $n$ girls sitting around a circular table, where $n>3$. In every move, we are allowed to swap
the places of $2$ adjacent children. The [b]entropy[/b] of a configuration is the minimal number of moves
such that at the end of them each child has at least one neighbor of the same gender.
Find the maximal possible entropy over the set of all configurations.
[i]Authored by Viktor Simjanoski[/i]
2015 Purple Comet Problems, 1
Arvin ate 11 halves of tarts, Bernice ate 12 quarters of tarts, Chrisandra ate 13 eighths of tarts, and Drake ate 14 sixteenths of tarts. How many tarts were eaten?
1992 India Regional Mathematical Olympiad, 1
Determine the set of integers $n$ for which $n^2+19n+92$ is a square.
2006 AMC 10, 19
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?
$ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 59 \qquad \textbf{(D) } 89 \qquad \textbf{(E) } 178$
2005 All-Russian Olympiad, 1
Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?
III Soros Olympiad 1996 - 97 (Russia), 9.8
Some lottery is played as follows. A lottery participant buys a card with $10$ numbered cells. He has the right to cross out any $4$ of these $10$ cells. Then a drawing occurs, during which some $7$ out of $10$ cells become winning. The player wins when all $4$ squares he crosses out are winning. The question arises, what is the smallest number of cards that can be used so that, if filled out correctly, at least one of these cards will win in any case? We do not suggest that you answer this question (we ourselves do not know the answer), although, of course, we will be very glad if you do and will evaluate this achievement accordingly. The task is; to indicate a certain number $n$ and a method of filling n cards that guarantees at least one win. The smaller $n$, the higher the rating of the work.
2013 Argentina National Olympiad Level 2, 2
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
1986 IMO Longlists, 51
Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.
2024 Israel TST, P2
In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.
2015 Iran Team Selection Test, 6
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2010 Princeton University Math Competition, 8
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
2009 Croatia Team Selection Test, 2
On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.
1989 Cono Sur Olympiad, 2
Let $ABCD$ be a square with diagonals $AC$ and $BD$, and $P$ a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.
2010 Serbia National Math Olympiad, 1
Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that
\[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\]
Find all $n$ for which equality can be attained.
[i]Proposed by Aleksandar Ilic[/i]
2010 Princeton University Math Competition, 1
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?
2008 Balkan MO Shortlist, G8
Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$
2007 Nicolae Coculescu, 3
Show that for any three numbers $ a,b,c\in (1,\infty ) , $ the following inequality is true:
$$ \log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab$$
[i]Costel Anghel[/i]
Estonia Open Junior - geometry, 1997.1.3
Juku invented an apparatus that can divide any segment into three equal segments. How can you find the midpoint of any segment, using only the Juku made, a ruler and pencil?
1975 Miklós Schweitzer, 3
Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$.
[i]L. Megyesi[/i]
2009 Harvard-MIT Mathematics Tournament, 4
How many functions $f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}$ satisfy $f(f(x)) = f(x)$ for all $x\in\{ 1,2, 3, 4, 5\}$?
2024 AMC 10, 25
Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$[sup]th[/sup] brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$?
[img]https://cdn.artofproblemsolving.com/attachments/2/d/b18d3d0a9e5005c889b34e79c6dab3aaefeffd.png[/img]
$
\textbf{(A) }88 \qquad
\textbf{(B) }89 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }91 \qquad
\textbf{(E) }92 \qquad
$
III Soros Olympiad 1996 - 97 (Russia), 11.2
Is there a function $f(x)$ defined and continuous on $R$ such that:
a) $f(f(x)) = 1 + 2x$ ?
b) $f(f(x)) = 1 - 2x $?
2012 Indonesia TST, 2
Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.
2015 HMNT, 9
Rosencrantz plays $n \leq 2015$ games of question, and ends up with a win rate $\left(\text{i.e.}\: \frac{\text{\# of games won}}{\text{\# of games played}}\right)$ of $k$. Guildenstern has also played several games, and has a win rate less than $k$. He realizes that if, after playing some more games, his win rate becomes higher than $k$, then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of $k$.