Found problems: 85335
Define a $ k$-[i]clique[/i] to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
[i]Proposed by Ray Li[/i]
Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night?
[i]Proposed by Espen Slettnes[/i]
Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?
Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, and radii $r_1$ and $r_2$, respectively. Suppose that $O_2$ is on $\omega_1$. Let $A$ be one of the intersections of $\omega_1$ and $\omega_2$, and $B$ be one of the two intersections of line $O_1O_2$ with $\omega_2$. If $AB = O_1A$, find all possible values of $\frac{r_1}{r_2}$
.
Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.
Let $F$ be a point inside a convex pentagon $ABCDE$, and let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ denote the distances from $F$ to the lines $AB$, $BC$, $CD$, $DE$, $EA$, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ are chosen on the inner bisectors of the angles $A$, $B$, $C$, $D$, $E$ of the pentagon respectively, so that $AF_{1} = AF$ , $BF_{2} = BF$ , $CF_{3} = CF$ , $DF_{4} = DF$ and $EF_{5} = EF$ . If the distances from $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ to the lines $EA$, $AB$, $BC$, $CD$, $DE$ are $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$, $b_{5}$, respectively.
Prove that $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}$
Let $I$ be the incenter of the scalene $\Delta ABC$, such, $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove, that,
$\text{(i) } \frac{AI}{IE}=\frac{ID}{DE}$
$\text{(ii) } IA=IF$
Assume that $n\ge 3$ people with different names sit around a round table. We call any unordered pair of them, say $M,N$, dominating if
1) they do not sit in adjacent seats
2) on one or both arcs connecting $M,N$ along the table, all people have names coming alphabetically after $M,N$.
Determine the minimal number of dominating pairs.
Let $n \ge 2$ be a positive integer. A grasshopper is moving along the sides of an $n \times n$ square net, which is divided on $n^2$ unit squares. It moves so that
а) in every $1 \times 1$ unit square of the net, it passes only through one side
b) when it passes one side of $1 \times1$ unit square of the net, it jumps on a vertex on another arbitrary $1 \times 1$ unit square of the net, which does not have a side on which the grasshopper moved along.
The grasshopper moves until the conditions can be fulfilled.
What is the shortest and the longest path that the grasshopper can go through if it moves according to the condition of the problem? Calculate its length and draw it on the net.
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used?
$\text{(A)}\ 6000 \qquad \text{(B)}\ 7500 \qquad \text{(C)}\ 24,000 \qquad \text{(D)}\ 30,000 \qquad \text{(E)}\ 37,500$
The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
\[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\]
When does equality hold?
[b]Q4.[/b] A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total length of each typr of road is the same, what is the average speed of his journey?
\[(A) \; 2 \text{km/h} \qquad (B) \; 2,5 \text{km/h} ; \qquad (C ) \; 3 \text{km/h} ; \qquad (D) \; 3,5 \text{km/h} ; \qquad (E) \; 4 \text{km/h}.\]
By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 11\qquad
\textbf{(E)}\ 12$
[u]Round 5[/u]
[b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$.
[b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. )
[b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$?
[u]Round 6[/u]
[b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$?
[b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally)
[b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$.
[u]Round 7[/u]
[b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit?
[b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$.
[b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$.
[u]Round 8[/u]
[b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of
people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of
times the letter c appears in the Speed Round. Estimate
$$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n \ne 0$ be a natural number. A sequence of numbers is briefly called a sequence “$F_n$” if $n$ different numbers $z_1$, $z_2$, $...$, $z_n$ exist so that the following conditions are fulfilled:
(1) Each term of the sequence is one of the numbers $z_1$, $z_2$, $...$, $z_n$.
(2) Each of the numbers $z_1$, $z_2$, $...$, $z_n$ occurs at least once in the sequence.
(3) Any two immediately consecutive members of the sequence are different numbers.
(4) No subsequence of the sequence has the form $\{a, b, a, b\}$ with $a \ne b$.
Note: A subsequence of a given sequence $\{x_1, x_2, x_3, ...\}$ or $\{x_1, x_2, x_3, ..., x_s\}$ is called any sequence of the form $\{x_{m1}, x_{m2}, x_{m3}, ...\}$ or $\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}$ with natural numbers $m_1 < m_2 < m_3 < ...$
Answer the following questions:
a) Given $n$, are there sequences $F_n$ of arbitrarily long length?
b) If question (a) is answered in the negative for an $n$:
What is the largest possible number of terms that a sequence $F_n$ can have (given $n$)?
Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$.
Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.
Chords $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$ of a sphere meet at an interior point $P$ but are not contained in a plane. The sphere through $A$, $B$, $C$, $P$ is tangent to the sphere through $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $P$. Prove that $\, AA' = BB' = CC'$.
Let $a_1,a_2,\cdots$ be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers $\mathbb{N}$ into infinitely many subsets with infinite cardinality $A_1,A_2,\cdots$, so that for every subset $A_i$, if we denote $b_1<b_2<\cdots$ be the elements of $A_i$, then for every $k\in \mathbb{N}$ and for every $1\le i\le a_k$, it satisfies $b_{i+1}-b_{i}\le k$?
Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$.
Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$
Do there exist distinct reals $x, y, z$, such that $\frac{1}{x^2+x+1}+\frac{1}{y^2+y+1}+\frac{1}{z^2+z+1}=4$?
The numbers $2,4,\ldots,2^{100}$ are written on a board. At a move, one may erase the numbers $a,b$ from the board and replace them with $ab/(a+b).$ Prove that the last numer on the board will be greater than 1.
[i]From the folklore[/i]
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.