This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2015 Saudi Arabia Pre-TST, 1.4
Tags: table , combinatorics , coloring
We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

1997 Taiwan National Olympiad, 9
Tags: combinatorics , logarithm
For $n\geq k\geq 3$, let $X=\{1,2,...,n\}$ and let $F_{k}$ a the family of $k$-element subsets of $X$, any two of which have at most $k-2$ elements in common. Show that there exists a subset $M_{k}$ of $X$ with at least $[\log_{2}{n}]+1$ elements containing no subset in $F_{k}$.

2023 Costa Rica - Final Round, 3.1
Tags: functional equation , algebra , nonnegative integers
Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\] Determine the value of $f(2023)/f(2022)$.

MOAA Team Rounds, 2023.7
Tags:
In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

2013 AMC 12/AHSME, 20
Tags: trigonometry , geometry , trapezoid , parabola , analytic geometry , graphing lines , conic
For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$? $ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $

2011 Junior Balkan Team Selection Tests - Moldova, 2
Tags: inequalities , algebra
The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$. Prove that $| x | \le15$.

1952 AMC 12/AHSME, 27
Tags: ratio , geometry , perimeter
The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

2012 May Olympiad, 1
Tags: algebra , number theory
Pablo says: “I add $2$ to my birthday and multiply the result by $2$. I add to the number obtained $4$ and multiply the result by $5$. To the new number obtained I add the number of the month of my birthday (for example, if it's June, I add $6$) and I get $342$. " What is Pablo's birthday date? Give all the possibilities

2004 France Team Selection Test, 2
Tags: trigonometry , angle bisector , geometry
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

2020 Brazil Cono Sur TST, 1
Tags: geometry , incenter , circumcircle
Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

1994 China National Olympiad, 2
Tags: pigeonhole principle , combinatorics
There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$). An [i]operation[/i] is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite [i]operations[/i].

2024 ELMO Shortlist, N5
Tags: number theory
Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

1994 National High School Mathematics League, 9
Tags:
Point Sets $A=\{(x,y)|(x-3)^2+(y-4)^2\leq\left( \frac{5}{2}\right)^2\},B=\{(x,y)|(x-4)^2+(y-5)^2>\left( \frac{5}{2}\right)^2\}$, then the number of integral points in $A\cap B$ is________.

2017 CCA Math Bonanza, I1
Tags: factorial
Find the integer $n$ such that $6!\times7!=n!$. [i]2017 CCA Math Bonanza Individual Round #1[/i]

2005 Kyiv Mathematical Festival, 1
Tags: inequalities
Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.

1999 VJIMC, Problem 1
Tags: limit , real analysis
Find the limit $$\lim_{n\to\infty}\left(\prod_{k=1}^n\frac k{k+n}\right)^{e^{\frac{1999}n}-1}.$$

2022 MMATHS, 1
Tags: algebra
Suppose that $ a + b = 20$, $b + c = 22$, and $c + a = 2022$. Compute $\frac{a-b}{c-a}$ .

2014 Macedonia National Olympiad, 2
Tags: number theory
Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

2021 Austrian MO National Competition, 6
Tags: perfect square , number theory
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

1978 IMO Longlists, 15
Tags: number theory
Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.

2023 CUBRMC, 3
Tags: number theory
Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.

2022 Caucasus Mathematical Olympiad, 8
Tags: binomial coefficient , algebra
Paul can write polynomial $(x+1)^n$, expand and simplify it, and after that change every coefficient by its reciprocal. For example if $n=3$ Paul gets $(x+1)^3=x^3+3x^2+3x+1$ and then $x^3+\frac13x^2+\frac13x+1$. Prove that Paul can choose $n$ for which the sum of Paul’s polynomial coefficients is less than $2.022$.

1975 Spain Mathematical Olympiad, 4
Tags: algebra , inequalities , sum , product
Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

2023 LMT Fall, 14
Tags: geometry
In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.) [i]Proposed by Jonathan Liu[/i]

2000 Moldova Team Selection Test, 5
Tags: number theory
Let $(F_n)_{n\in\mathbb{N}}$ be the Fibonacci sequence difined as $F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}$. Show that for every nonnegative integer $r$ there is a term in the Fibonacci sequence that is divided by $r$.