Found problems: 85335
2020 Durer Math Competition Finals, 8
The integers $1, 2, 3, 4, 5$ and $6$ are written on a board. You can perform the following kind of move: select two of the numbers, say $a$ and $b$, such that $4a - 2b$ is nonnegative; erase $a$ and $b$, then write down $4a - 2b$ on the board (hence replacing two of the numbers by just one). Continue performing such moves until only one number remains on the board. What is the smallest possible positive value of this last remaining number?
2023 Belarusian National Olympiad, 11.7
Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $AB$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$
Find all possible values of $\frac{AB+AC}{BC}$
2020-21 KVS IOQM India, 26
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$. Find this sum.
1978 Chisinau City MO, 154
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?
2019 MIG, 6
Square $ABCD$ has side length $4$. Side $AB$ is extended to point $E$ so that $AE$ has the same length as $AC$, as shown below. What is the length of $EC$? Express your answer as a decimal to the nearest hundredth.
[asy]
size(80);
defaultpen(fontsize(8pt));
pair EE = (4sqrt(2),0);
pair A = (0,0);
pair B = (4,0);
pair C = (4,4);
pair D = (0,4);
draw(A--B--C--D--cycle);
draw(A--EE);
draw(C--EE,dotted);
label("$A$",A,SW);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("$E$",EE,S);
[/asy]
2010 Romania Team Selection Test, 4
Two circles in the plane, $\gamma_1$ and $\gamma_2$, meet at points $M$ and $N$. Let $A$ be a point on $\gamma_1$, and let $D$ be a point on $\gamma_2$. The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$, respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$, respectively. Assume the order $M$, $N$, $F$, $A$, $E$ is circular around $\gamma_1$, and the segments $AB$ and $DE$ are congruent. Prove that the points $A$, $F$, $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above.
[i]***[/i]
2024 India National Olympiad, 5
Let points $A_1$, $A_2$ and $A_3$ lie on the circle $\Gamma$ in a counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1,2,3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of rotation is equial to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))$, where the indices are taken modulo $3$ (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$).
Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$.
[i]Proposed by Anant Mudgal[/i]
2004 Thailand Mathematical Olympiad, 17
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$.
2022 Stanford Mathematics Tournament, 9
Let $f(x,y)=(\cos x+y\sin x)^2$. We may express $\text{max}_xf(x,y)$, the maximum value of $f(x,y)$ over all values of $x$ for a given fixed value of $y$, as a function of $y$, call it $g(y)$. Let the smallest positive value $x$ which achieves this maximum value of $f(x,y)$ for a given $y$ be $h(y)$. Compute
\[\int_1^{2+\sqrt{3}}\frac{h(y)}{g(y)}\text{d}y.\]
2024 LMT Fall, A1
In Genshin Impact, $PRIMOGEM'$ is the octagon in the diagram below. Let $A$ be the intersection of $PO$ and $IE$. Suppose $PR=RI=IM=MO=OG=GE=EM'=M'P$, $AP=AI=AO=AE=4$, and $AR=AM=AG=AM'=\sqrt{2}$. Find the area of $PRIMOGEM'$.
[asy]
size(5cm);
pair P = (0, 4), R = (1, 1), I = (4, 0), M = (1, -1), O = (0, -4), G = (-1, -1), E = (-4, 0), MM = (-1, 1), origin = (0, 0);
draw(P--R--I--M--O--G--E--MM--P);
draw(origin--P);
draw(origin--I);
draw(origin--O);
draw(origin--E);
draw(R--G);
draw(MM--M);
label("$P$", P, N);
label("$R$", R, NE);
label("$I$", I, SE);
label("$M$", M, SE);
label("$G$", G, SW);
label("$E$", E, W);
label("$M'$", MM, NW);
label("$O$", O, S);
[/asy]
2015 Balkan MO Shortlist, A1
If ${a, b}$ and $c$ are positive real numbers, prove that
\begin{align*}
a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}
[i](Montenegro).[/i]
2011 Pre-Preparation Course Examination, 5
suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent:
[b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$
[b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).
2011 Hanoi Open Mathematics Competitions, 3
What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ?
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.
2014 District Olympiad, 4
Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.
2024 CMIMC Integration Bee, 2
\[\int_0^2 |\sin(\pi x)|+|\cos(\pi x)|\mathrm dx\]
[i]Proposed by Anagh Sangavarapu[/i]
2020 Czech-Austrian-Polish-Slovak Match, 1
Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear.
(Patrik Bak, Slovakia)
2017 Turkey EGMO TST, 2
At the beginning there are $2017$ marbles in each of $1000$ boxes. On each move Aybike chooses a box, grabs some of the marbles from that box and delivers them one for each to the boxes she wishes. At least how many moves does Aybike have to make to have different number of marbles in each box?
2011 NIMO Summer Contest, 11
How many ordered pairs of positive integers $(m, n)$ satisfy the system
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2,
\\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
1981 Poland - Second Round, 5
In the plane there are two disjoint sets $ A $ and $ B $, each of which consists of $ n $ points, and no three points of the set $ A \cup B $ lie on one straight line. Prove that there is a set of $ n $ disjoint closed segments, each of which has one end in the set $ A $ and the other in the set $ B $.
2025 CMIMC Algebra/NT, 1
Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?
2016 KOSOVO TST, 4
$f:R->R$ such that :
$f(1)=1$ and for any $x\in R$
i) $f(x+5)\geq f(x)+5$
ii)$f(x+1)\leq f(x)+1$
If $g(x)=f(x)+1-x$ find g(2016)
2016 Junior Regional Olympiad - FBH, 2
If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$
PEN S Problems, 32
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?
2014 Belarusian National Olympiad, 3
The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.
2018 Purple Comet Problems, 18
Rectangle $ABCD$ has side lengths $AB = 6\sqrt3$ and $BC = 8\sqrt3$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.