Found problems: 85335
2009 AMC 10, 15
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
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(-2,2)--(-3,3);
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label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
2001 Turkey MO (2nd round), 1
Find all ordered triples of positive integers $(x,y,z)$ such that
\[3^{x}+11^{y}=z^{2}\]
1962 AMC 12/AHSME, 33
The set of $ x$-values satisfying the inequality $ 2 \leq |x\minus{}1| \leq 5$ is:
$ \textbf{(A)}\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad
\textbf{(B)}\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad
\textbf{(C)}\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad
\textbf{(D)}\ \minus{}1 \leq x \leq 3 \qquad
\textbf{(E)}\ \minus{}4 \leq x \leq 6$
2022 Israel TST, 3
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.
Russian TST 2022, P1
A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots.
[i]Proposed by P. Kozhevnikov[/i]
2007 Hanoi Open Mathematics Competitions, 6
Let $P(x) = x^3 + ax^2 + bx + 1$ and $|P(x)| \leq 1$ for all x such that $|x| \leq 1$.
Prove that $|a| + |b| \leq 5$.
2017-IMOC, C1
On a blackboard , the 2016 numbers $\frac{1}{2016} , \frac{2}{2016} ,... \frac{2016}{2016}$ are written.
One can perfurm the following operation : Choose any numbers in the blackboard, say $a$ and$ b$ and replace them by $2ab-a-b+1$.
After doing 2015 operation , there will only be one number $t$ Onthe blackboard .
Find all possible values of $ t$.
2022 Brazil Team Selection Test, 3
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right.
Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors.
[i]Proposed by Gurgen Asatryan, Armenia[/i]
1955 Miklós Schweitzer, 10
[b]10.[/b] Show that if a convex polyhedron has vertices of regular distribution and congruent faces, then it is regular. (A system of points is said to be of regular distribution if every point of the system can be transformed into any other point by congruent transformations mapping the system onto itself.) [b](G. 11)[/b]
2017 MMATHS, 4
In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.
1984 IMO Longlists, 22
In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$
2007 Tuymaada Olympiad, 2
Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?
2023 Germany Team Selection Test, 1
In a triangle $\triangle ABC$ with orthocenter $H$, let $BH$ and $CH$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. If the tangent line to the circumcircle of $\triangle ABC$ passing through $A$ intersects $BC$ at $P$, $M$ is the midpoint of $AH$, and $EF$ intersects $BC$ at $G$, then prove that $PM$ is parallel to $GH$.
[i]Proposed by Sreejato Bhattacharya[/i]
2023 Taiwan TST Round 3, 5
Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy:
(1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards.
(2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards.
Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$.
[i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$.
[i]Proposed by usjl[/i]
1963 Putnam, A1
i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon.
ii) Let $P_1 , P_2 ,\ldots, P_{12}$ be the vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.
2013 Kyiv Mathematical Festival, 2
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?
1967 IMO Longlists, 11
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
LMT Accuracy Rounds, 2022 S2
Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.
2022 JHMT HS, 10
Let $R$ be the rectangle in the coordinate plane with corners $(0, 0)$, $(20, 0)$, $(20, 22)$, and $(0, 22)$, and partition $R$ into a $20\times 22$ grid of unit squares. For a given line in the coordinate plane, let its [i]pixelation[/i] be the set of grid squares in $R$ that contain part of the line in their interior. If $P$ is a point chosen uniformly at random in $R$, then compute the expected number of sets of grid squares that are pixelations of some line through $P$.
2000 Austrian-Polish Competition, 3
For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$
1992 AMC 12/AHSME, 11
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is
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[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $
2018 CMIMC Combinatorics, 9
Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.
2021 Serbia Team Selection Test, P3
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.
2023 BMT, 1
Compute the three-digit number that satisfies the following properties:
$\bullet$ The hundreds digit and ones digit are the same, but the tens digit is different.
$\bullet$ The number is divisible by $9$.
$\bullet$ When the number is divided by $5$, the remainder is $1$.
2012 Turkey MO (2nd round), 1
Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$