Found problems: 85335
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by [i]Nikola Velov, Macedonia[/i]
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is [i]monotonically bounded[/i] if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a [i]mountain[/i] if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
[i]Proposed by Michael Ren[/i]
Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .
a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.
b) Show that $A$ is an infinite set.
A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$.
[i]Proposed by S. Berlov[/i]
For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that
$$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.
Let $f:\mathbb{R} \to \mathbb{R}$, such that
i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$, exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$
ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$, exists $x,y \in \mathbb{R}$ with $ b - \epsilon < x < b < y < b + \epsilon$, such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$
Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$.
Consider the set $S$ of lattice points $(x,y)$ with $0\le x,y\le 8$. Call a function $f:S\to \{1,2,\dots, 9\}$ a [i]Sudoku function[/i] if:
[list]
[*] $\{ f(x,0), f(x,1), \dots, f(x,8)\} = \{1,2,\dots, 9\}$ for each $0\le x\le 8$ and $\{ f(0,y), f(1,y), \dots, f(8,y) \} = \{1,2,\dots, 9\}$ for each $0\le y\le 8$.
[*] For any integers $0\le m,n\le 2$ and any $0\le i_1,j_1,i_2,j_2\le 2$, $f(3m+i_1, 3n+j_1)\neq f(3m+i_2, 3n+j_2)$ unless $i_1=i_2$ and $j_1=j_2$.
[/list]
Over all Sudoku functions $f$, compute the maximum possible value of $\sum_{0\le i\le 8} f(i,i) + \sum_{0\le i\le 7} f(i, i+1)$.
[i]Proposed by Brandon Wang[/i]
We say that a positive integer $n$ is [i]good[/i] if the result of multiplying the first $n$ positive odd integers consists only of the digits $1$, $3$, $5$ and $9$. For example, $n = 3$ is good because $1 \times 3 \times 5 = 15$, but $n = 4$ is not good because $1 \times 3 \times 5 \times 7 = 105$. Determine all the good numbers.
In acute-angled triangle $ ABC$, the points $ D,E,F$ are on sides $ BC,CA,AB$, respectively such that $ \angle AFE \equal{} \angle BFD, \angle FDB \equal{} \angle EDC, \angle DEC \equal{} \angle FEA$. Prove that $ AD$ is perpendicular to $ BC$.
For any finite sequence of positive integers $\pi$, let $S(\pi)$ be the number of strictly increasing sub sequences in $\pi$ with length $2$ or more. For example, in the sequence $\pi = \{3, 1, 2, 4\}$, there are five increasing sub-sequences: $\{3, 4\}$, $\{1, 2\}$, $\{1, 4\}$, $\{2, 4\}$, and \${1, 2, 4\}, so $S(\pi) = 5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order $\pi$ from left to right in her hand. Determine
$$\sum_{\pi} S(\pi),$$
where the sum is taken over all possible orders $\pi$ of the card values.
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]
One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$
At a festival, Jing Jing plays a game where she must knock down ten targets with as few balls as possible. Every time Jing Jing knocks down a target, she can reuse the ball she just threw and does not have to pick up a new ball. Suppose that Jing Jing knocks down each target with a probability of $\tfrac{3}{4}$. Compute the expected number of balls that Jing Jing needs to knock down all ten targets.
Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds:
$$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$
[i]Proposed by Zaza Meliqidze, Georgia[/i]
Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the midpoint of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$.
a) Prove that all the possible lines $(KP)$ pass through one point.
b) Find the set of all the possible points $P$.
If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$?
$\textbf{(A)}\ 27\qquad
\textbf{(B)}\ 29 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 105\qquad
\textbf{(E)}\ 127$
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white.
Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.
Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
$ \textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$