Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$. Determine the size of the set $\{ \det A : A \in M \}$. [i]Here $\det A$ denotes the determinant of the matrix $A$.[/i]

At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are [i]already [/i] in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.

Prove that in all triangles $\vartriangle ABC$ with $\angle A = 2 \angle B$ it holds that, if $D$ is the foot of the perpendicular from $C$ to the perpendicular bisector of $AB$, $\frac{AC}{DC}$ is constant for any value of $\angle B$.

Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\frac{1}{y},y+\frac{1}{z}$ and $z+\frac{1}{x}$ are integers.

In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$. The experimenter wishes to announce a value for $C$ in which every digit is significant. That is, whatever $C$ is, the announced value must be the correct result when C is rounded to that number of digits. The most accurate value the experimenter can announce for $C$ is $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2.4\qquad\textbf{(C)}\ 2.43\qquad\textbf{(D)}\ 2.44\qquad\textbf{(E)}\ 2.439 $

Find all function $f:\mathbb{R}\to \mathbb{R}$ such that \[f(x)f(y)=f(xy-1)+yf(x)+xf(y)\] for all $x,y \in \mathbb{R}$

Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$, $$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$. (a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$. (b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$. (c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.

Find the total number of integer solutions of the equation $$x^5-y^2=4$$ [i](Erken)[/i]

Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

[b]p1.[/b] An Egyptian fraction has the form $1/n$, where $n$ is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, $3/5$ was seen as $1/2 + 1/10$, or $1/3 + 1/4 + 1/60$. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for $3/5$ would be $1/2 + 1/10$. a) Find the greedy Egyptian fraction representations for $2/13$. b) Find the greedy Egyptian fraction representations for $9/10$. c) Find the greedy Egyptian fraction representations for $2/(2k+1)$, where $k$ is a positive integer. d) Find the greedy Egyptian fraction representations for $3/(6k+1)$, where $k$ is a positive integer. [b]p2.[/b] a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii. [img]https://cdn.artofproblemsolving.com/attachments/8/1/7c4d81ebfbd4445dc31fa038d9dc68baddb424.png[/img] b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number $d$. The area of the larger triangle is twice the area of the smaller triangle. Find $d$. [img]https://cdn.artofproblemsolving.com/attachments/8/7/1f0d56d8e9e42574053c831fa129eb40c093d9.png[/img] [b]p3.[/b] Suppose that the domain of a function $f$ is the set of real numbers and that $f$ takes values in the set of real numbers. A real number $x_0$ is a fixed point of f if $f(x_0) = x_0$. a) Let $f(x) = m x + b$. For which $m$ does $f$ have a fixed point? b) Find the fixed point of f$(x) = m x + b$ in terms of m and b, when it exists. c) Consider the functions $f_c(x) = x^2 - c$. i. For which values of $c$ are there two different fixed points? ii. For which values of $c$ are there no fixed points? iii. In terms of $c$, find the value(s) of the fixed point(s). d) Find an example of a function that has exactly three fixed points. [b]p4.[/b] A square based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level. a) How many balls are there in the pyramid? b) If each ball has a radius of $1$ meter, how tall is the pyramid? c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls? [b]p5.[/b] We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly $0$) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of $4$ suits, $13$ denominations, and $0$ jokers. a) For a deck with $3$ suits $\{a, b, c\}$ and $7$ denominations $\{1, 2, 3, 4, 5, 6, 7\}$, and $0$ jokers, find the probability that a 3-card hand will be a straight. (A straight consists of $3$ cards in sequence, e.g., $1 \heartsuit$ ,$2 \spadesuit$ , $3\clubsuit$ , $2\diamondsuit$ but not $6 \heartsuit$ ,$7 \spadesuit$ , $1\diamondsuit$). b) For a deck with $3$ suits, $7$ denominations, and $0$ jokers, find the probability that a $3$-card hand will consist of $3$ cards of the same suit (i.e., a flush). c) For a deck with $3$ suits, $7$ denominations, and $1$ joker, find the probability that a $3$-card hand dealt at random will be a straight and also the probability that a $3$-card hand will be a flush. d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain $1$ joker and is to have identical probabilities for a straight and a flush when a $3$-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For what real numbers $p$ has the system of equations $$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$ just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?

Find the sum of the reciprocals of the positive integral factors of $84$.

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction? [img]https://wiki-images.artofproblemsolving.com//thumb/a/a2/2024_AMC_8_-19.png/1200px-2024_AMC_8_-19.png[/img] $\textbf{(A) } 0\qquad\textbf{(B) } \dfrac{1}{5} \qquad\textbf{(C) } \dfrac{4}{15} \qquad\textbf{(D) } \dfrac{1}{3} \qquad\textbf{(E) } \dfrac{2}{5}$

Consider the polynomial $$P\left(t\right)=t^3-29t^2+212t-399.$$ Find the product of all positive integers $n$ such that $P\left(n\right)$ is the sum of the digits of $n$.

Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that $f(km)+f(kn)-f(k)f(mn)\ge 1$ for all $k,m,n\in\mathbb{N}$.

Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that: \[ w^{2p}+x^{2p}+y^{2p}=z^{2p}. \]

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of \[ x^2 \plus{} px \plus{} q \equal{} 0 \] to be $ r^2$ and $ s^2$, $ p$ must equal: $ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad \textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad \textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\ \textbf{(D)}\ B^2 \minus{} 2C\qquad \textbf{(E)}\ 2C \minus{} B^2$

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that \[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]

Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? $\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$

Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$ has perimeter $k$ units and area $k$ square units for some real number $k$.

For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\}$ $=$ $x-$ $\lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying $$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$ If $S$ is the sume of all numbers in $A$, find $\lfloor S \rfloor$

Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.