Found problems: 85335
2009 239 Open Mathematical Olympiad, 2
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.
1999 BAMO, 2
Let $O = (0,0), A = (0,a), and B = (0,b)$, where $0<b<a$ are reals. Let $\Gamma$ be a circle with diameter $\overline{AB}$ and let $P$ be any other point on $\Gamma$. Line $PA$ meets the x-axis again at $Q$. Prove that angle $\angle BQP = \angle BOP$.
2013 AMC 10, 20
A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?
$ \textbf{(A)}\ 1-\frac{\sqrt2}2+\frac\pi4\qquad\textbf{(B)}\ \frac12+\frac\pi4\qquad\textbf{(C)}\ 2-\sqrt2+\frac\pi4\qquad\textbf{(D)}\ \frac{\sqrt2}2+\frac\pi4\qquad\textbf{(E)}\ 1+\frac{\sqrt2}4+\frac\pi8 $
1979 Canada National Olympiad, 5
A walk consists of a sequence of steps of length 1 taken in the directions north, south, east, or west. A walk is self-avoiding if it never passes through the same point twice. Let $f(n)$ be the number of $n$-step self-avoiding walks which begin at the origin. Compute $f(1)$, $f(2)$, $f(3)$, $f(4)$, and show that
\[2^n < f(n) \le 4 \cdot 3^{n - 1}.\]
1957 Kurschak Competition, 3
What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?
2018 Iran Team Selection Test, 2
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).
At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?
[i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]
2016 BMT Spring, 14
Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?
2015 Olympic Revenge, 2
Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.
1961 AMC 12/AHSME, 31
In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $
1999 AMC 12/AHSME, 11
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $ 1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $ 9$ and four centers to label locker number $ 10$. If it costs $ \$137.94$ to label all the lockers, how many lockers are there at the school?
$ \textbf{(A)}\ 2001 \qquad
\textbf{(B)}\ 2010 \qquad
\textbf{(C)}\ 2100 \qquad
\textbf{(D)}\ 2726 \qquad
\textbf{(E)}\ 6897$
2024/2025 TOURNAMENT OF TOWNS, P2
There are $N$ pupils in a school class, and there are several communities among them. Sociability of a pupil will mean the number of pupils in the largest community to which the pupil belongs (if the pupil belongs to none then the sociability equals $1$). It occurred that all girls in the class have different sociabilities. What is the maximum possible number of girls in the class?
1998 China Team Selection Test, 3
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2019 Taiwan TST Round 3, 2
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
1978 Miklós Schweitzer, 7
Let $ T$ be a surjective mapping of the hyperbolic plane onto itself which maps collinear points into collinear points. Prove that $ T$ must be an isometry.
[i]M. Bognar[/i]
1966 Miklós Schweitzer, 8
Prove that in Euclidean ring $ R$ the quotient and remainder are always uniquely determined if and only if $ R$ is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: $ R$ can also be a field or the null ring.)
[i]E. Fried[/i]
1954 AMC 12/AHSME, 7
A housewife saved $ \$2.50$ in buying a dress on sale. If she spent $ \$25$ for the dress, she saved about:
$ \textbf{(A)}\ 8 \% \qquad
\textbf{(B)}\ 9 \% \qquad
\textbf{(C)}\ 10 \% \qquad
\textbf{(D)}\ 11 \% \qquad
\textbf{(E)}\ 12 \%$
2012 Today's Calculation Of Integral, 779
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane.
When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.
2019 China Girls Math Olympiad, 8
For a tournament with $8$ vertices, if from any vertex it is impossible to follow a route to return to itself, we call the graph a [i]good[/i] graph. Otherwise, we call it a [i]bad[/i] graph. Prove that
$(1)$ there exists a tournament with $8$ vertices such that after changing the orientation of any at most $7$ edges of the tournament, the graph is always a[i]bad[/i] graph;
$(2)$ for any tournament with $8$ vertices, one can change the orientation of at most $8$ edges of the tournament to get a [i]good[/i] graph.
(A tournament is a complete graph with directed edges.)
1986 Miklós Schweitzer, 10
Let $X_1, X_2$ be independent, identically distributed random variables such that $X_i\geq 0$ for all $i$. Let $\mathrm EX_i=m$, $\mathrm{Var} (X_i)=\sigma ^2<\infty$. Show that, for all $0<\alpha\leq 1$
$$\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}$$
[Gy. Michaletzki]
1978 IMO Longlists, 3
Find all numbers $\alpha$ for which the equation
\[x^2 - 2x[x] + x -\alpha = 0\]
has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)
2022 MOAA, 15
Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, respectively. Let $O$ be the circumcenter of $ABC$. If $BI_B$ is perpendicular to $AO$, $AI_C = 3$ and $AC = 4\sqrt2$, then $AB^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Note: In triangle $\vartriangle ABC$, the $A$-excenter is the intersection of the exterior angle bisectors of $\angle ABC$ and $\angle ACB$. The $B$-excenter and $C$-excenter are defined similarly.
2004 Olympic Revenge, 2
If $a,b,c,x$ are positive reals, show that
$$\frac{a^{x+2}+1}{a^xbc+1}+\frac{b^{x+2}+1}{b^xac+1}+\frac{c^{x+2}+1}{c^xab+1}\geq 3$$
2011 AMC 8, 22
What is the tens digit of $7^{2011}$?
$ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}3\qquad\textbf{(D)}4\qquad\textbf{(E)}7 $
2024 China Western Mathematical Olympiad, 5
Given hexagon $ \mathcal{P}$ inscribed in a unit square, such that each vertex is on the side of the square. It’s known that all interior angles of the hexagon are equal. Find the maximum possible value of the smallest side length of $\mathcal{P}$.
2020 Dutch BxMO TST, 1
For an integer $n \ge 3$ we consider a circle with $n$ points on it.
We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other.
For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?