Found problems: 85335
2007 Today's Calculation Of Integral, 203
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]
2006 Macedonia National Olympiad, 4
Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ .
$(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$.
$(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.
1966 Miklós Schweitzer, 1
Show that a segment of length $ h$ can go through or be tangent to at most $ 2\lfloor h/\sqrt{2}\rfloor\plus{}2$ nonoverlapping unit
spheres.
[i]L.Fejes-Toth, A. Heppes[/i]
2004 Tournament Of Towns, 6
At the beginning of a two-player game, the number $2004!$ is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most $20$ different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains $0$. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?
2005 Purple Comet Problems, 3
Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy]
2024 District Olympiad, P3
Let $k$ be a positive integer. A ring $(A,+,\cdot)$ has property $P_k$ if for any $a,b\in A$ there exists $c\in A$ such that $a^k=b^k+c^k.$[list=a]
[*]Give an example of a finite ring $(A,+,\cdot)$ which [i]does not[/i] have $P_k$ for any $k\geqslant 2.$
[*]Let $n\geqslant 3$ be an integer and $M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}.$ Prove that all the elements of $M_n$ are odd integers and that $(M_n,\cdot)$ is a monoid.
[/list]
2017 Indonesia MO, 2
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:
[list]
[*]A shakes hands with B, and B shakes hands with C, or
[*]A doesn't shake hands with B, and B doesn't shake hands with C.
[/list]
If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.
2022 Princeton University Math Competition, A1 / B3
Given two polynomials $f$ and $g$ satisfying $f(x) \ge g(x)$ for all real $x,$ a [i]separating line[/i] between $f$ and $g$ is a line $h(x) = mx+k$ such that $f(x) \ge h(x) \ge g(x)$ for all real $x.$ Consider the set of all possible separating lines between $f(x) = x^2 - 2x + 5$ and $g(x) = 1 - x^2.$ The set of slopes of these lines is a closed interval $[a,b].$ Determine $a^4 + b^4.$
2017 Czech-Polish-Slovak Junior Match, 6
On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.
LMT Speed Rounds, 2011.8
There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron where you wait in a group with five other souls, each of which had already come through one of the entrances as well, to get a ride across. In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through
the entrances does not matter, and the entrance you went through is fixed.
1963 AMC 12/AHSME, 20
Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\dfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\dfrac{1}{4}$ miles per hour for the first hour, at $3\dfrac{3}{4}$ miles per hour for the second hour,
and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:
$\textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 2$
2015 Czech and Slovak Olympiad III A, 6
Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.
Kyiv City MO Juniors 2003+ geometry, 2022.8.3
In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$.
[i](Proposed by Danylo Khilko)[/i]
2023 OMpD, 4
Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$
1997 Slovenia National Olympiad, Problem 4
The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?
2018 CMIMC Team, 8-1/8-2
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.
2012 Argentina Cono Sur TST, 5
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.
2013 JBMO Shortlist, 3
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2019 Teodor Topan, 1
Do exist pairwise distinct matrices $ A,B,C\in \mathcal{M}_2(\mathbb{R}) $ verifying the following properties?
$ \text{(i)} \det A=\det C$
$ \text{(ii)} AB=C,BC=A,CA=B $
$ \text{(iii)} \text{tr} A,\text{tr} B\neq 0 $
[i]Robert Pop[/i]
1999 IberoAmerican, 1
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
2017 District Olympiad, 2
[b]a)[/b] Prove that there exist two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ having the properties:
$ \text{(i)}\quad f\circ g=g\circ f $
$\text{(ii)}\quad f\circ f=g\circ g $
$ \text{(iii)}\quad f(x)\neq g(x), \quad \forall x\in\mathbb{R} $
[b]b)[/b] Show that if there are two functions $ f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} $ with the properties $ \text{(i)} $ and $ \text{(iii)} $ from above, then $ \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , $ for all real numbers $ x. $
1894 Eotvos Mathematical Competition, 3
The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.
2006 USAMO, 6
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.
2018 Bangladesh Mathematical Olympiad, 5
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $E$ . The two circles passing through $B$ meet again at $F$ . The two circles passing through $C$ meet again at $G$. The two circles passing through $D$ meet again at $H$. Suppose, $ E, F, G,H $ are all distinct. Is the quadrilateral $EFGH$ similar to $ABCD$ ? Show with proof.
1977 AMC 12/AHSME, 13
If $a_1,a_2,a_3,\dots$ is a sequence of positive numbers such that $a_{n+2}=a_na_{n+1}$ for all positive integers $n$, then the sequence $a_1,a_2,a_3,\dots$ is a geometric progression
$\textbf{(A) }\text{for all positive values of }a_1\text{ and }a_2\qquad$
$\textbf{(B) }\text{if and only if }a_1=a_2\qquad$
$\textbf{(C) }\text{if and only if }a_1=1\qquad$
$\textbf{(D) }\text{if and only if }a_2=1\qquad $
$\textbf{(E) }\text{if and only if }a_1=a_2=1$