Found problems: 85335
2016 Tuymaada Olympiad, 4
Non-negative numbers $a$, $b$, $c$ satisfy
$a^2+b^2+c^2\geq 3$. Prove the inequality
$$
(a+b+c)^3\geq 9(ab+bc+ca).
$$
2006 Korea Junior Math Olympiad, 3
In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.
2013 Kosovo National Mathematical Olympiad, 1
Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.
1987 Greece National Olympiad, 4
Let $A,B$ be two points interior of circle $C(O,R)$ and $M$ a point on the circle. Let $A_1,B_1$ be the intersections of the circle with lines $MA$,$MB$ respectively. Let $G$ be the midpoint of $AB$and $G_1= C\cap MG$. Prove that$$\frac{MA}{AA_1}+ \frac{MB}{BB_1}> 2\frac{MG}{GG_1}$$
2021 Science ON all problems, 4
Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\
$\textbf{(a)}$ Prove that $G$ is cyclic. \\
$\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?
2014 NIMO Problems, 1
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
LMT Speed Rounds, 2017
[b]p1.[/b] Find the number of zeroes at the end of $20^{17}$.
[b]p2.[/b] Express $\frac{1}{\sqrt{20} +\sqrt{17}}$ in simplest radical form.
[b]p3.[/b] John draws a square $ABCD$. On side $AB$ he draws point $P$ so that $\frac{BP}{PA}=\frac{1}{20}$ and on side $BC$ he draws point $Q$ such that $\frac{BQ}{QC}=\frac{1}{17}$ . What is the ratio of the area of $\vartriangle PBQ$ to the area of $ABCD$?
[b]p4.[/b] Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints?
[b]p5.[/b] Alex is playing a game with an unfair coin which has a $\frac15$ chance of flipping heads and a $\frac45$ chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins?
[b]p6.[/b] Positive two-digit number $\overline{ab}$ has $8$ divisors. Find the number of divisors of the four-digit number $\overline{abab}$.
[b]p7.[/b] Call a positive integer $n$ diagonal if the number of diagonals of a convex $n$-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to $2017$.
[b]p8.[/b] There are $4$ houses on a street, with $2$ on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors.
[b]p9.[/b] Compute $$|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.$$
[b]p10.[/b] Given points $A,B$ in the coordinate plane, let $A \oplus B$ be the unique point $C$ such that $\overline{AC}$ is parallel to the $x$-axis and $\overline{BC}$ is parallel to the $y$-axis. Find the point $(x, y)$ such that $((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y)$.
[b]p11.[/b] In the following subtraction problem, different letters represent different nonzero digits.
$\begin{tabular}{ccccc}
& M & A & T & H \\
- & & H & A & M \\
\hline
& & L & M & T \\
\end{tabular}$
How many ways can the letters be assigned values to satisfy the subtraction problem?
[b]p12.[/b] If $m$ and $n$ are integers such that $17n +20m = 2017$, then what is the minimum possible value of $|m-n|$?
[b]p13. [/b]Let $f(x)=x^4-3x^3+2x^2+7x-9$. For some complex numbers $a,b,c,d$, it is true that $f (x) = (x^2+ax+b)(x^2+cx +d)$ for all complex numbers $x$. Find $\frac{a}{b}+ \frac{c}{d}$.
[b]p14.[/b] A positive integer is called an imposter if it can be expressed in the form $2^a +2^b$ where $a,b$ are non-negative integers and $a \ne b$. How many almost positive integers less than $2017$ are imposters?
[b]p15.[/b] Evaluate the infinite sum $$\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...$$
[b]p16.[/b] Each face of a regular tetrahedron is colored either red, green, or blue, each with probability $\frac13$ . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors?
[b]p17.[/b] Let $(k,\sqrt{k})$ be the point on the graph of $y=\sqrt{x}$ that is closest to the point $(2017,0)$. Find $k$.
[b]p18.[/b] Alice is going to place $2016$ rooks on a $2016 \times 2016$ chessboard where both the rows and columns are labelled $1$ to $2016$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations.
[b]p19.[/b] Let $f (n)$ be a function defined recursively across the natural numbers such that $f (1) = 1$ and $f (n) = n^{f (n-1)}$. Find the sum of all positive divisors less than or equal to $15$ of the number $f (7)-1$.
[b]p20.[/b] Find the number of ordered pairs of positive integers $(m,n)$ that satisfy
$$gcd \,(m,n)+ lcm \,(m,n) = 2017.$$
[b]p21.[/b] Let $\vartriangle ABC$ be a triangle. Let $M$ be the midpoint of $AB$ and let $P$ be the projection of $A$ onto $BC$. If $AB = 20$, and $BC = MC = 17$, compute $BP$.
[b]p22.[/b] For positive integers $n$, define the odd parent function, denoted $op(n)$, to be the greatest positive odd divisor of $n$. For example, $op(4) = 1$, $op(5) = 5$, and $op(6) =3$. Find $\sum^{256}_{i=1}op(i).$
[b]p23.[/b] Suppose $\vartriangle ABC$ has sidelengths $AB = 20$ and $AC = 17$. Let $X$ be a point inside $\vartriangle ABC$ such that $BX \perp CX$ and $AX \perp BC$. If $|BX^4 -CX^4|= 2017$, the compute the length of side $BC$.
[b]p24.[/b] How many ways can some squares be colored black in a $6 \times 6$ grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct.
[b]p25.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = 2$, $AD = 4$, and $\angle ABC = 120^o$. Let $M$ be the midpoint of $BD$. If $\angle AMC = 90^o$, find the length of segment $CD$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2016 Hanoi Open Mathematics Competitions, 13
Let $H$ be orthocenter of the triangle $ABC$. Let $d_1, d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB, AC$ at $D, E$ and the line d_2 intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$ if and only if $F$ is the midpoint of segment $BC$.
2001 Italy TST, 4
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
2019 Moldova Team Selection Test, 10
The circle $\Omega$ with center $O$ is circumscribed to acute triangle $ABC$. Let $P$ be a point on the circumscribed circle of $OBC$, such that $P$ is inside $ABC$ and is different from $B$ and $C$. Bisectors of angles $BPA$ and $CPA$ intersect the sides $AB$ and $AC$ in points $E$ and $F.$ Prove that the incenters of triangles $PEF, PCA$ and $PBA$ are collinear.
1984 AMC 12/AHSME, 30
For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then
\[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \]
equals
$\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$
2016 Junior Balkan Team Selection Tests - Romania, 3
Let $n$ be an integer greater than $2$ and consider the set
\begin{align*}
A = \{2^n-1,3^n-1,\dots,(n-1)^n-1\}.
\end{align*}
Given that $n$ does not divide any element of $A$, prove that $n$ is a square-free number. Does it necessarily follow that $n$ is a prime?
2003 All-Russian Olympiad, 3
Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$
2019 Mid-Michigan MO, 7-9
[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions.
[b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale.
[b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number?
[b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it?
[b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get?
[img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 Durer Math Competition (First Round), 3
Let $n \ge 3$ be an integer and $A$ be a subset of the real numbers of size n. Denote by $B$ the set of real numbers that are of the form $ x \cdot y$, where $x, y \in A$ and $x\ne y$. At most how many distinct positive primes could $B$ contain (depending on $n$)?
2003 Brazil National Olympiad, 2
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.
2009 Indonesia TST, 4
Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.
2015 Grand Duchy of Lithuania, 2
Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.
2013 USAJMO, 1
Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?
Kvant 2024, M2788
An equilateral triangle $\mathcal{T}{}$ with side 111 is divided by straight lines parallel to its sides into equilateral triangles with side 1. The vertices of these small triangles, except the centre of $\mathcal{T}{}$ are marked. Call a set of several marked points [i]linear[/i] if[list=i][*]the marked points lie on a line $\ell$ parallel to one of the sides of the triangle $\mathcal{T}$ and;
[*]if two marked points on $\ell$ are in this set, every other marked point inbetween them is in the set.
[/list]How many ways are there to split all the marked points into 111 linear sets?
2024 Austrian MO National Competition, 1
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
2005 Moldova Team Selection Test, 4
Find the largest positive $p$ ($p>1$) such, that $\forall a,b,c\in[\frac1p,p]$ the following inequality takes place
\[9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4\]
2017 Iberoamerican, 6
Let $n > 2$ be an even positive integer and let $a_1 < a_2 < \dots < a_n$ be real numbers such that $a_{k + 1} - a_k \leq 1$ for each $1 \leq k \leq n - 1$. Let $A$ be the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is even, and let $B$ the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is odd. Show that
$$\prod_{(i, j) \in A} (a_j - a_i) > \prod_{(i, j) \in B} (a_j - a_i)$$
2009 Cuba MO, 2
Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.
2001 India IMO Training Camp, 2
Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.