Found problems: 85335
Kvant 2023, M2766
Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student
[list]
[*]the number of selected cells on each row, and
[*]that there is one selected cell on each column.
[/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?
[i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]
2020 Harvard-MIT Mathematics Tournament, 9
Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$.
[i]Proposed by Andrew Gu.[/i]
2023 ELMO Shortlist, A2
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
[i]Proposed by Luke Robitaille[/i]
2017 CCA Math Bonanza, T1
Given that $9\times10\times11\times\cdots\times15=32432400$, what is $1\times3\times5\times\cdots\times15$?
[i]2017 CCA Math Bonanza Team Round #1[/i]
1992 IMO Longlists, 25
[b][i](a) [/i][/b] Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$, and $C$ satisfying the following conditions:
\[A^2 = A, B^2 = C, C^2 = B,\] \[AB = B, AC = C, BC = A,\]
where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$, and $H^2$ denotes $HH.$
[b][i](b)[/i][/b] Show that for every such partition of $\mathbb N$, $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$
1979 IMO Shortlist, 18
Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$
2020-2021 OMMC, 3
Two real numbers $x, y$ are chosen randomly and independently on the interval $(1, r)$ where $r$ is some real number between $1024$ and $2048$. Let $P$ be the probability that $\lfloor \log_2x \rfloor > \lfloor \log_2y \rfloor .$ The value of $P$ is maximized when $r = \frac{p}{q}$ where $p,q$ are relatively prime positive integers. Find $p+q.$
2004 All-Russian Olympiad Regional Round, 10.4
$N \ge 3$ different points are marked on the plane. It is known that among pairwise distances between marked points there are not more than $n$ different distances. Prove that $N \le (n + 1)^2$.
2007 Junior Balkan Team Selection Tests - Romania, 3
Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.
Gheorghe Țițeica 2024, P3
Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible.
[i]Vlad Matei[/i]
2018 APMO, 4
Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.
2020 LMT Fall, 32
In a lottery there are $14$ balls, numbered from $1$ to $14$. Four of these balls are drawn at random. D'Angelo wins the lottery if he can split the four balls into two disjoint pairs, where the two balls in each pair have difference at least $5$. The probability that D'Angelo wins the lottery can be expressed as $\frac{m}{n}$, with $m,n$ relatively prime. Find $m+n$.
[i]Proposed by Richard Chen[/i]
2007 Gheorghe Vranceanu, 3
$ \lim_{n\to\infty } \sqrt[n]{\sum_{i=0}^n\binom{n}{i}^2} $
2011 NIMO Summer Contest, 4
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality
\[
1 < a < b+2 < 10.
\]
[i]Proposed by Lewis Chen
[/i]
Russian TST 2015, P1
Find all pairs of natural numbers $(a,b)$ satisfying the following conditions:
[list]
[*]$b-1$ is divisible by $a+1$ and
[*]$a^2+a+2$ is divisible by $b$.
[/list]
1959 AMC 12/AHSME, 10
In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Then $\overline{AE}$ is:
$ \textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6 $
2016 Online Math Open Problems, 12
For each positive integer $n\ge 2$, define $k\left(n\right)$ to be the largest integer $m$ such that $\left(n!\right)^m$ divides $2016!$. What is the minimum possible value of $n+k\left(n\right)$?
[i]Proposed by Tristan Shin[/i]
2007 Harvard-MIT Mathematics Tournament, 35
[i]The Algorithm.[/i] There are thirteen broken computers situated at the following set $S$ of thirteen points in the plane:
\[\begin{array}{ccc}A=(1,10)&B=(976,9)&C=(666,87)\\D=(377,422)&E=(535,488)&F=(775,488) \\ G=(941,500) & H=(225,583)&I=(388,696)\\J=(3,713)&K=(504,872)&L=(560,934)\\&M=(22,997)&\end{array}\]
At time $t=0$, a repairman begins moving from one computer to the next, traveling continuously in straight lines at unit speed. Assuming the repairman begins and $A$ and fixes computers instantly, what path does he take to minimize the [i]total downtime[/i] of the computers? List the points he visits in order. Your score will be $\left\lfloor \dfrac{N}{40}\right\rfloor$, where \[N=1000+\lfloor\text{the optimal downtime}\rfloor - \lfloor \text{your downtime}\rfloor ,\] or $0$, whichever is greater. By total downtime we mean the sum \[\sum_{P\in S}t_P,\] where $t_P$ is the time at which the repairman reaches $P$.
1973 IMO Shortlist, 12
Consider the two square matrices
\[A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1 \end{bmatrix} \quad \text{ and } \quad B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}\]
with entries $+1$ and $-1$. The following operations will be called elementary:
(1) Changing signs of all numbers in one row;
(2) Changing signs of all numbers in one column;
(3) Interchanging two rows (two rows exchange their positions);
(4) Interchanging two columns.
Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.
2005 Tournament of Towns, 4
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always $10$ cm. Each side of the table is more than $1$ meter long. At the initial moment both ants are on the same side of the table.
(a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?
(b) [i](4 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
VII Soros Olympiad 2000 - 01, 10.5
An acute-angled triangle $ABC$ is given. Points $A_1, B_1$ and $C_1$ are taken on its sides $BC, CA$ and $AB$, respectively, such that
$\angle B_1A_1C_1 + 2 \angle BAC = 180^o$,
$\angle A_1C_1B_1 + 2 \angle ACB = 180^o$,
$\angle C_1B_1A_1 + 2 \angle CBA = 180^o$.
Find the locus of the centers of the circles inscribed in triangles $A_1B_1C_1$ (all kinds of such triangles are considered).
2016 Hong Kong TST, 5
Let $ABCD$ be inscribed in a circle with center $O$. Let $E$ be the intersection of $AC$ and $BD$. $M$ and $N$ are the midpoints of the arcs $AB$ and $CD$ respectively (the arcs not containing any other vertices). Let $P$ be the intersection point of $EO$ and $MN$. Suppose $BC=5$, $AC=11$, $BD=12$, and $AD=10$. Find $\frac{MN}{NP}$
Today's calculation of integrals, 897
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
2023 AMC 12/AHSME, 4
Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$?
$\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$