Found problems: 85335
2016 Mediterranean Mathematics Olympiad, 2
Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that
\[ \sqrt{\frac{b}{a^2+3}}+
\sqrt{\frac{c}{b^2+3}}+
\sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]
2017 Polish MO Finals, 4
Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.
1946 Putnam, A6
A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$, where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.
2007 District Olympiad, 4
Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$
Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:
2018 Tuymaada Olympiad, 5
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
[i]Proposed by N. Vlasova, S. Berlov[/i]
1991 Spain Mathematical Olympiad, 5
For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.
2023 China MO, 5
Prove that there exist $C>0$, which satisfies the following conclusion:
For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree.
Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$.
[i]Proposed by Qu Zhenhua[/i]
2015 Iran MO (2nd Round), 2
There's a special computer and it has a memory. At first, it's memory just contains $x$.
We fill up the memory with the following rules.
1) If $f\neq 0$ is in the memory, then we can also put $\frac{1}{f}$ in it.
2) If $f,g$ are in the memory, then we can also put $ f+g$ and $f-g$ in it.
Find all natural number $n$ such that we can have $x^n$ in the memory.
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]
1981 Spain Mathematical Olympiad, 1
Calculate the sum of $n$ addends
$$7 + 77 + 777 +...+ 7... 7.$$
1998 National High School Mathematics League, 4
Statement $P$: solution set to inequalities $a_1x^2+b_1x+c_1>0$ and $a_2x^2+b_2x+c_2>0$ are the same; statement $Q$: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
$\text{(A)}$ $Q$ is sufficient and necessary condition of $P$.
$\text{(B)}$ $Q$ is sufficient but unnecessary condition of $P$.
$\text{(C)}$ $Q$ is insufficient but necessary condition of $P$.
$\text{(D)}$ $Q$ is insufficient and unnecessary condition of $P$.
2017 Purple Comet Problems, 30
A container is shaped like a right circular cone with base diameter $18$ and height $12$. The vertex of the container is pointing down, and the container is open at the top. Four spheres, each with radius $3$, are placed inside the container as shown. The fi rst sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are placed so they are each tangent to the cone and tangent to the rst sphere, and the second and fourth spheres are each tangent to the third sphere. The volume of the tetrahedron whose vertices are at the centers of the spheres is $K$. Find $K^2$.
[img]https://cdn.artofproblemsolving.com/attachments/9/c/648ec2cf0f0c2f023cd00b1c0595a9396d0ddc.png[/img]
2014 Dutch BxMO/EGMO TST, 3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
1985 IMO Shortlist, 11
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
2016 Saudi Arabia BMO TST, 2
Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.
2018 AMC 8, 22
Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$
[asy]
size(5cm);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((0,6)--(6,0)); draw((3,0)--(6,6));
label("$A$",(0,6),NW);
label("$B$",(6,6),NE);
label("$C$",(6,0),SE);
label("$D$",(0,0),SW);
label("$E$",(3,0),S);
label("$F$",(4,2),E);
[/asy]
$\textbf{(A) } 100 \qquad \textbf{(B) } 108 \qquad \textbf{(C) } 120 \qquad \textbf{(D) } 135 \qquad \textbf{(E) } 144$
2019 Romania Team Selection Test, 2
Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
2023 Bulgaria National Olympiad, 5
For every positive integer $n$ determine the least possible value of the expression
\[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\]
given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$.
2010 Bosnia and Herzegovina Junior BMO TST, 4
On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?
2001 Tournament Of Towns, 2
At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.
2022 Dutch IMO TST, 4
Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively.
Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.
VMEO III 2006 Shortlist, A2
Given a polynomial $P(x)=x^4+3x^2-9x+1$. Calculate $P(\alpha^2+\alpha+1)$ where\[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]
2014 National Olympiad First Round, 13
Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{ADB} \right)=15^{\circ}$, $m \left (\widehat{BCD} \right)=90^{\circ}$. The diagonals of quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC|=4, |EA|=8$ and $|EP|=2$. What is $m \left (\widehat{PBD} \right)$?
$
\textbf{(A)}\ 15^{\circ}
\qquad\textbf{(B)}\ 30^{\circ}
\qquad\textbf{(C)}\ 45^{\circ}
\qquad\textbf{(D)}\ 60^{\circ}
\qquad\textbf{(E)}\ 75^{\circ}
$
2025 Benelux, 3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.
2010 All-Russian Olympiad Regional Round, 10.4
We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.