Found problems: 85335
1979 IMO Longlists, 29
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.
2022 Junior Balkan Team Selection Tests - Moldova, 3
Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A straight line is drawn through point $B$, which again intersects circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Point $E$, located on circle $\omega_1$ , satisfies the relation $CE = CB$ , and point $F$, located on circle $\omega_2$, satisfies the relation $DB = DF$. The line $BF$ intersects again the circle $\omega_1$ at the point $P$, and the line $BE$ intersects again the circle $\omega_2$ at the point $Q$. Prove that the points $A, P$, and $Q$ are collinear.
2021 MOAA, 16
Let $1,7,19,\ldots$ be the sequence of numbers such that for all integers $n\ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence.
[i]Proposed by Nathan Xiong[/i]
2020 Brazil Cono Sur TST, 1
Determine the quantity of positive integers $N$ of $10$ digits with the following properties:
I- All the digits of $N$ are non-zero.
II- $11|N$.
III- $N$ and all the permutation(s) of the digits of $N$ are divisible by $12$.
1975 Putnam, A5
Let $I\subset \mathbb{R}$ be an interval and $f(x)$ a continuous real-valued function on $I$. Let $y_1$ and $y_2$ be linearly independent solutions of $y''=f(x)y$ taking positive values on $I$. Show that for some positive number $k$ the function $k\cdot\sqrt{y_1 y_2}$ is a solution of $y''+\frac{1}{y^{3}}=f(x)y$.
2019 Chile National Olympiad, 4
In the convex quadrilateral $ABCD$ , $\angle ADC = \angle BCD > 90^o$ . Let $E$ be the intersection of the line $AC$ with the line parallel to $AD$ that passes through $B$. Let $F$ be the intersection of line $BD$ with the line parallel to $BC$ passing through $A$. Prove that $EF$ is parallel to $CD$.
2004 IMO Shortlist, 8
Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$.
[i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]
2014 ASDAN Math Tournament, 15
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?
2015 BMT Spring, 3
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.
.
2021 China Girls Math Olympiad, 7
In an acute triangle $ABC$, $AB \neq AC$, $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$. $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$. Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$.
Proof that $B, C, Y, O, Z$ lie on a circle.
2017 USAJMO, 1
Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
2019 China Northern MO, 4
A manager of a company has 8 workers. One day, he holds a few meetings.
(1)Each meeting lasts 1 hour, no break between two meetings.
(2)Three workers attend each meeting.
(3)Any two workers have attended at least one common meeting.
(4)Any worker cannot leave until he finishes all his meetings.
Then, how long does the worker who works the longest work at least?
2005 India IMO Training Camp, 3
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
2008 AMC 8, 1
Susan had $\$50$ to spend at the carnival. She spent $\$12$ on food and twice as much on rides. How many dollars did she have left to spend?
$\textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 26 \qquad
\textbf{(D)}\ 38 \qquad
\textbf{(E)}\ 50 $
2016 Czech-Polish-Slovak Junior Match, 1
Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$.
Czech Republic
2017-IMOC, C3
Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?
2025 USA IMO Team Selection Test, 2
Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and
\begin{align*}
a_{n+1} &= a_n^2 - 2b_n\\
b_{n+1} &= b_n^2 - 2a_n
\end{align*}
for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing (that is, there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k > N$).
[i]Holden Mui[/i]
2007 Swedish Mathematical Competition, 1
Solve the following system
\[
\left\{ \begin{array}{l}
xyzu-x^3=9 \\
x+yz=\dfrac{3}{2}u \\
\end{array} \right.
\]
in positive integers $x$, $y$, $z$ and $u$.
1992 Bulgaria National Olympiad, Problem 4
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]
2021 Macedonian Mathematical Olympiad, Problem 2
In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.
2021 CCA Math Bonanza, L2.1
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. The line through $A$ perpendicular to $AC$ intersects line $BC$ at $D$, and the line through $C$ perpendicular to $AC$ intersects line $AB$ at $E$. Compute the area of triangle $BDE$.
[i]2021 CCA Math Bonanza Lightning Round #2.1[/i]
1988 AMC 12/AHSME, 18
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives 5th prize and the winner bowls #3 in another game. The loser of this game receives 4th prize and the winner bowls #2. The loser of this game receives 3rd prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ \text{none of these} $
2018 India Regional Mathematical Olympiad, 4
Suppose $100$ points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?
2020 Macedonian Nationаl Olympiad, 2
Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that
$|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$,
with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.
2008 SDMO (Middle School), 3
In the diagram, $AD:DB=1:1$, $BE:EC=1:2$, and $CF:FA=1:3$. If the area of triangle $ABC$ is $120$, then find the area of triangle $DEF$.
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