Found problems: 85335
2012 Romania National Olympiad, 2
Let $ABC$ be a triangle with right $\angle A$. Consider points $D \in (AC)$ and $E \in (BD)$ such that $\angle ABC = \angle ECD = \angle CED$. Prove that $BE = 2 \cdot AD$
2005 May Olympiad, 3
A segment $AB$ of length $100$ is divided into $100$ little segments of length $1$ by $99$ intermediate points. Endpoint $A$ is assigned $0$ and endpoint $B$ is assigned $1$. Gustavo assigns each of the $99$ intermediate points a $0$ or a $1$, at his choice, and then color each segment of length $1$ blue or red, respecting the following rule:
[i]The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. [/i] Determine if Gustavo can assign the $0$'s and $1$'s so as to get exactly $30$ blue segments. And $35$ blue segments? (In each case, if the answer is yes, show a distribution of $0$'s and $1$'s, and if the answer is no, explain why).
2018 Harvard-MIT Mathematics Tournament, 7
A tourist is learning an incorrect way to sort a permutation $(p_1, \dots, p_n)$ of the integers $(1, \dots, n)$. We define a [i]fix[/i] on two adjacent elements $p_i$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_i>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a=1, 2, \dots, n-1$. In round $a$ of fixes, the tourist fixes $p_a$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on, up to $p_{n-1}$ and $p_n$. In this process, there are $(n-1)+(n-2)+\dots+1 = \tfrac{n(n-1)}{2}$ total fixes performed. How many permutations of $(1, \dots, 2018)$ can the tourist start with to obtain $(1, \dots, 2018)$ after performing these steps?
2021 Polish MO Finals, 4
Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$,
$$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$
2007 Hong kong National Olympiad, 4
find all positive integer pairs $(m,n)$,satisfies:
(1)$gcd(m,n)=1$,and $m\le\ 2007$
(2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$
2017 AMC 10, 7
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
$\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$
1994 Miklós Schweitzer, 8
Prove that a Hausdorff space X is countably compact iff for every open cover $\cal {U}$ there is a finite set $A \subset X$ such that $ \bigcup \{U \in {\cal U} : U \cap A \neq \emptyset \} = X$.
2022 Bulgaria JBMO TST, 3
For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$.
(I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)
2013 Czech-Polish-Slovak Junior Match, 2
Each positive integer should be colored red or green in such a way that the following two conditions are met:
- Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red.
- Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green.
Determine all such colorings.
1998 AMC 12/AHSME, 25
A piece of graph paper is folded once so that $ (0,2)$ is matched with $ (4,0)$ and $ (7,3)$ is matched with $ (m,n)$. Find $ m \plus{} n$.
$ \textbf{(A)}\ 6.7\qquad
\textbf{(B)}\ 6.8\qquad
\textbf{(C)}\ 6.9\qquad
\textbf{(D)}\ 7.0\qquad
\textbf{(E)}\ 8.0$
2021 Alibaba Global Math Competition, 16
Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that
\[\text{dim} V^{H_1}=\text{dim} V^{H_2},\]
where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that
\[Z(G) \cap H_1=Z(G) \cap H_2.\]
Here $Z(G)$ denotes the center of $G$.
2004 Thailand Mathematical Olympiad, 12
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
2011 QEDMO 8th, 5
$9$ points are given in the interior of the unit square.
Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points.
2012 Today's Calculation Of Integral, 795
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$
2003 AMC 12-AHSME, 2
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $ \$4$ per pair and each T-shirt costs $ \$5$ more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $ \$2366$, how many members are in the League?
$ \textbf{(A)}\ 77 \qquad
\textbf{(B)}\ 91 \qquad
\textbf{(C)}\ 143 \qquad
\textbf{(D)}\ 182 \qquad
\textbf{(E)}\ 286$
2000 France Team Selection Test, 1
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.
2015 Greece National Olympiad, 3
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$.
A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$,
B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$
2005 Iran MO (2nd round), 1
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
2015 Online Math Open Problems, 15
Let $a$, $b$, $c$, and $d$ be positive real numbers such that
\[a^2 + b^2 - c^2 - d^2 = 0 \quad \text{and} \quad a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad).\]
Let $M$ be the maximum possible value of $\tfrac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $100m + n$.
[i]Proposed by Robin Park[/i]
1996 Canadian Open Math Challenge, 6
In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.
2004 China Second Round Olympiad, 2
In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$. The x-intercept of line $A_nB_n$ is $a_n$, and the x-coordinate of point $B_n$ is $b_n$, $n\in\mathbb{N}$. Prove that
(1) $a_n>a_{n+1}>4$, $n\in\mathbb{N}$;
(2) There is $n_0\in\mathbb{N}$, such that for any $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.
2014 Online Math Open Problems, 30
For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m$. Define an [i]$n$-plant[/i] to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it [i]perfect[/i] if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.
Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Yang Liu[/i]
2013 European Mathematical Cup, 1
In each field of a table there is a real number. We call such $n \times n$ table [i]silly[/i] if each entry equals the product of all the numbers in the neighbouring fields.
a) Find all $2 \times 2$ silly tables.
b) Find all $3 \times 3$ silly tables.
1983 AIME Problems, 4
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.
[asy]
size(150); defaultpen(linewidth(0.65)+fontsize(11));
real r=10;
pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C;
path P=circle(O,r);
C=intersectionpoint(B--(B.x+r,B.y),P);
draw(Arc(O, r, 45, 360-17.0312));
draw(A--B--C);dot(A); dot(B); dot(C);
label("$A$",A,NE);
label("$B$",B,SW);
label("$C$",C,SE);
[/asy]
1993 Irish Math Olympiad, 2
Let $ a_i,b_i$ $ (i\equal{}1,2,...,n)$ be real numbers such that the $ a_i$ are distinct, and suppose that there is a real number $ \alpha$ such that the product $ (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n)$ is equal to $ \alpha$ for each $ i$. Prove that there is a real number $ \beta$ such that $ (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j)$ is equal to $ \beta$ for each $ j$.