Found problems: 85335
2015 Dutch IMO TST, 4
Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.
2002 AMC 8, 6
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. One of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time. Which one is it?
[asy]
size(450);
defaultpen(linewidth(0.8));
path[] p={origin--(8,8)--(14,8), (0,10)--(4,10)--(14,0), origin--(14,14), (0,14)--(14,14), origin--(7,7)--(14,0)};
int i;
for(i=0; i<5; i=i+1) {
draw(shift(21i,0)*((0,16)--origin--(14,0)));
draw(shift(21i,0)*(p[i]));
label("Time", (7+21i,0), S);
label(rotate(90)*"Volume", (21i,8), W);
}
label("$A$", (0*21 + 7,-5), S);
label("$B$", (1*21 + 7,-5), S);
label("$C$", (2*21 + 7,-5), S);
label("$D$", (3*21 + 7,-5), S);
label("$E$", (4*21 + 7,-5), S);
[/asy]
$\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$
2018 Pan African, 1
Find all functions $f : \mathbb Z \to \mathbb Z$ such that $$(f(x + y))^2 = f(x^2) + f(y^2)$$ for all $x, y \in \mathbb Z$.
2022 AMC 12/AHSME, 22
Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following?
$\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$
2001 National Olympiad First Round, 8
Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ -14
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2022 Princeton University Math Competition, B1
A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees.
1996 AMC 8, 17
Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates $(2,2)$. What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?
[asy]
pair O,P,Q,R,T;
O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0);
draw((-5,0)--(3,0)); draw((0,-1)--(0,3));
draw(P--Q--R);
draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8));
draw((-0.2,2.8)--(0,3)--(0.2,2.8));
draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2));
draw((2.8,-0.2)--(3,0)--(2.8,0.2));
draw(Q--T);
label("$O$",O,SW);
label("$P$",P,S);
label("$Q$",Q,NE);
label("$R$",R,W);
label("$T$",T,S);
[/asy]
NOT TO SCALE
$\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)$
2014 Harvard-MIT Mathematics Tournament, 24
Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.
2016 Harvard-MIT Mathematics Tournament, 5
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom 92 = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
2020 Brazil Team Selection Test, 1
Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.
1986 AMC 12/AHSME, 11
In $\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. The length of $HM$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair H=origin, A=(0,6), B=(-4,0), C=(5,0), M=B+3.6*dir(B--A);
draw(B--C--A--B^^M--H--A^^rightanglemark(A,H,C));
label("$A$", A, NE);
label("$B$", B, W);
label("$C$", C, E);
label("$H$", H, S);
label("$M$", M, dir(M));
[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 7.5\qquad\textbf{(E)}\ 8 $
2024 Bulgaria MO Regional Round, 12.2
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
2021 Cono Sur Olympiad, 5
Given an integer $n \geq 3$, determine if there are $n$ integers $b_1, b_2, \dots , b_n$, distinct two-by-two (that is, $b_i \neq b_j$ for all $i \neq j$) and a polynomial $P(x)$ with coefficients integers, such that $P(b_1) = b_2, P(b_2) = b_3, \dots , P(b_{n-1}) = b_n$ and $P(b_n) = b_1$.
1982 IMO Longlists, 55
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.
1985 Greece National Olympiad, 3
Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.
2024 Argentina National Olympiad Level 2, 6
A list of $7$ numbers is constructed using the following procedure: each number in the list is equal to the sum of the previous number and the previous number written in reverse order. For example, if a number in the list is $23544$, the next number is $68076 = 23544 + 44532$. (It is forbidden for any number in the list to start with $0$, although the reversed numbers may start with $0$.) Decide whether it is possible to choose the first number of the list so that the seventh number is a prime number.
2015 Sharygin Geometry Olympiad, 1
Circles $\alpha$ and $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.
(D. Mukhin)
2015 Greece Team Selection Test, 2
Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$
1996 Poland - Second Round, 6
Prove that every interior point of a parallelepiped with edges $a,b,c$ is on the distance at most $\frac12 \sqrt{a^2 +b^2 +c^2}$ from some vertex of the parallelepiped.
2008 Chile National Olympiad, 3
Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$
2004 National Olympiad First Round, 32
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
2020 Tournament Of Towns, 2
Three legendary knights are fighting against a multiheaded dragon.
Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads?
Alexey Zaslavsky
2003 Romania Team Selection Test, 8
Two circles $\omega_1$ and $\omega_2$ with radii $r_1$ and $r_2$, $r_2>r_1$, are externally tangent. The line $t_1$ is tangent to the circles $\omega_1$ and $\omega_2$ at points $A$ and $D$ respectively. The parallel line $t_2$ to the line $t_1$ is tangent to the circle $\omega_1$ and intersects the circle $\omega_2$ at points $E$ and $F$. The line $t_3$ passing through $D$ intersects the line $t_2$ and the circle $\omega_2$ in $B$ and $C$ respectively, both different of $E$ and $F$ respectively. Prove that the circumcircle of the triangle $ABC$ is tangent to the line $t_1$.
[i]Dinu Serbanescu[/i]
2003 Tournament Of Towns, 4
A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).
A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.
2019 ELMO Shortlist, C4
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]