Found problems: 85335
2019 Dutch IMO TST, 3
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
2005 District Olympiad, 3
Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.
2016 BMT Spring, 2
Cyclic quadrilateral $ABCD$ has side lengths $AB = 6$, $BC = 7$, $CD = 7$, $DA = 6$. What is the area of $ABCD$?
2017 Math Prize for Girls Problems, 4
If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.
2022 Balkan MO Shortlist, C4
Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus.
[i]Proposed by Tommy Walker Mackay, United Kingdom[/i]
2011 Putnam, A3
Find a real number $c$ and a positive number $L$ for which
\[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]
2011 Purple Comet Problems, 9
There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$
2016 Hanoi Open Mathematics Competitions, 11
Let $I$ be the incenter of triangle $ABC$ and $\omega$ be its circumcircle. Let the line $AI$ intersect $\omega$ at point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac12 \angle BAC$ . Let $X$ be the second point of intersection of line $EI$ with $\omega$ and $T$ be the point of intersection of segment $DX$ with line $AF$ . Prove that $TF \cdot AD = ID \cdot AT$ .
2010 LMT, 32
Compute the infinite sum $\frac{1^3}{2^1}+\frac{2^3}{2^2}+\frac{3^3}{2^3}+\dots+\frac{n^3}{2^n}+\dots.$
2004 Thailand Mathematical Olympiad, 13
Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.
2024 Nigerian MO Round 2, Problem 2
Solve the system of equations:
\[x>y>z\]
\[x+y+z=1\]
\[x^2+y^2+z^2=69\]
\[x^3+y^3+z^3=271\]
[hide=Answer]x=7, y=-2, z=-4[/hide]
2023 HMNT, 10
It is midnight on April $29$th, and Abigail is listening to a song by her favorite artist while staring at her clock, which has an hour, minute, and second hand. These hands move continuously. Between two consecutive midnights, compute the number of times the hour, minute, and second hands form two equal angles and no two hands overlap.
2013 AMC 8, 17
The sum of six consecutive positive integers is 2013. What is the largest of these six integers?
$\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$
Estonia Open Senior - geometry, 2020.1.5
A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$ . The line $BC$ intersects the circle $c$ for second time at point $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.
1976 IMO Shortlist, 3
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2002 Romania National Olympiad, 4
Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that:
$a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$;
$b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.
Putnam 1938, B4
The parabola $P$ has focus a distance $m$ from the directrix. The chord $AB$ is normal to $P$ at $A.$ What is the minimum length for $AB?$
1950 Kurschak Competition, 2
Three circles $C_1$, $C_2$, $C_3$ in the plane touch each other (in three different points). Connect the common point of $C_1$ and $C_2$ with the other two common points by straight lines. Show that these lines meet $C_3$ in diametrically opposite points.
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
$\text{(A)}\ (-6,0) \qquad \text{(B)}\ (-4,0) \qquad \text{(C)}\ (-2,0) \qquad \text{(D)}\ (2,0) \qquad \text{(E)}\ (4,0)$