Found problems: 85335
2023 BMT, 8
A circle intersects equilateral triangle $\vartriangle XY Z$ at $A,$ $B$, $C$, $D$, $E$, and $F$ such that points $X$, $A$, $B$, $Y$ , $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. If $AC^2 +CE^2 +EA^2 = 1900$ and $BD^2 + DF^2 + FB^2 = 2092$, compute the positive difference between the areas of triangles $\vartriangle ACE$ and $\vartriangle BDF$.
2017 QEDMO 15th, 11
Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.
1998 Gauss, 17
Claire takes a square piece of paper and folds it in half four times without unfolding, making an
isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the
paper would look like
2014 Belarus Team Selection Test, 3
$N$ cells are marked on an $n\times n$ table so that at least one marked cel is among any four cells of the table which form the figure [img]https://cdn.artofproblemsolving.com/attachments/2/2/090c32eb52df31eb81b9a86c63610e4d6531eb.png[/img] (tbe figure may be rotated). Find the smallest possible value of $N$.
(E. Barabanov)
2017 F = ma, 18
A uniform disk is being pulled by a force F through a string attached to its center of mass. Assume
that the disk is rolling smoothly without slipping. At a certain instant of time, in which region of the
disk (if any) is there a point moving with zero total acceleration?
A Region I
B Region II
C Region III
D Region IV
E All points on the disk have a nonzero acceleration
2018 Latvia Baltic Way TST, P15
Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations
$$\begin{cases}
x+y+z=n\\
xyz = 2t^3.
\end{cases}$$
2012 Today's Calculation Of Integral, 793
Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.
VI Soros Olympiad 1999 - 2000 (Russia), 8.6
Two players take turns writing down all proper non-decreasing fractions with denominators from $1 $ to $1999$ and at the same time writing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?
1996 Bundeswettbewerb Mathematik, 1
Can a square of side length $5$ be covered by three squares of side length $4$?
2019 Saudi Arabia JBMO TST, 2
Two circles, having their centers in A and B, intersect at points M and N. The radii AP and BQ are parallel and are in different semi-planes determined of the line AB. If the external common tangent intersect AB in D, and PQ intersects AB at C, prove that the <CND is right.
2022 Malaysia IMONST 2, 2
The following list shows every number for which more than half of its digits are digits $2$, in increasing order:
$$2, 22, 122, 202, 212, 220, 221, 222, 223, 224, \dots$$
If the $n$th term in the list is $2022$, what is $n$?
2016 Harvard-MIT Mathematics Tournament, 9
Fix positive integers $r>s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r-1$ that shares at least $s$ elements with each set in $F$.
2009 Postal Coaching, 3
Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.
PEN E Problems, 20
Verify that, for each $r \ge 1$, there are infinitely many primes $p$ with $p \equiv 1 \; \pmod{2^r}$.
1997 China National Olympiad, 1
Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions:
i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$;
ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ .
Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .
2023 Baltic Way, 10
On a circle, $n \geq 3$ points are marked. Each marked point is coloured red, green or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all $n$ for which there exists an initial state of $n$ marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.
1991 Tournament Of Towns, (296) 3
The numbers $x_1,x_2,x_3, ..., x_n$ satisfy the two conditions
$$\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1$$
Prove that there are two numbers among them whose product is no greater than $- 1/n$.
(Stolov, Kharkov)
1964 AMC 12/AHSME, 9
A jobber buys an article at $\$24$ less $12\frac{1}{2}\%$. He then wishes to sell the article at a gain of $33\frac{1}{3}\%$ of his cost after allowing a $20\%$ discount on his marked price. At what price, in dollars, should the article be marked?
${{ \textbf{(A)}\ 25.20 \qquad\textbf{(B)}\ 30.00 \qquad\textbf{(C)}\ 33.60 \qquad\textbf{(D)}\ 40.00 }\qquad\textbf{(E)}\ \text{none of these} } $
2020 CHMMC Winter (2020-21), 6
Anna and Bob are playing a game on a rectangular board with $i$ rows and $j$ columns. Anna and Bob alternate turns with Anna going first. On each turn, a player places a penny in a square and then all squares in the same row and column of that square are marked. A player cannot place a penny in any marked square. When a player cannot place a penny in any square, they lose and the other player wins. How many ordered pairs of integers $(i, j)$ with $1 \le i \le 2020, 1 \le j \le 2020$ are there such that Anna wins?
2001 Federal Math Competition of S&M, Problem 3
Let $k$ be a positive integer and $N_k$ be the number of sequences of length $2001$, all members of which are elements of the set $\{0,1,2,\ldots,2k+1\}$, and the number of zeroes among these is odd. Find the greatest power of $2$ which divides $N_k$.
2022 Princeton University Math Competition, B1
Let $a, b, c, d$ be real numbers for which $a^2 + b^2 + c^2 + d^2 = 1$. Show the following inequality:
$$a^2 + b^2 - c^2 - d^2 \le \sqrt{2 + 4(ac + bd)}.$$
2006 Baltic Way, 4
Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of
$\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$
and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.
2009 Canadian Mathematical Olympiad Qualification Repechage, 8
Determine an infinite family of quadruples $(a, b, c, d)$ of positive integers, each of which is a solution to $a^4+b^5+c^6=d^7$.
1992 China National Olympiad, 3
Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions:
1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ;
2) $2a_1=a_0+a_2-2$ ;
3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares.
Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.
2008 iTest Tournament of Champions, 5
Three circles with centers $V_0$, $V_1$, $V_2$ and radii $33$, $30$, $25$ respectively and mutually externally tangent: $P_i$ is the tangency point between circles $V_{i+1}$ and $V_{i+2}$, where indeces are taken modulo $3$. For $i=0,1,2$, line $P_{i+1}P_{i+2}$ intersects circle $V_{i+1}$ at $P_{i+2}$ and $Q_i$, and the same line intersects circle $V_{i+2}$ at $P_{i+1}$ and $R_i$. If $Q_0R_1$ intersects $Q_2R_0$ at $X$, then the distance from $X$ to line $R_1Q_2$ can be expressed as $\tfrac{a\sqrt b}c$, where the integer $b$ is not divisible by the square of any prime, and positive integers $a$ and $c$ are relatively prime. Find the value of $b+c$.