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

2000 Tuymaada Olympiad, 1
Tags: analytic geometry , topology , set theory , combinatorics
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?

2009 ISI B.Math Entrance Exam, 9
Tags: inequalities , triangle inequality , function
Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

2024 pOMA, 1
Tags: combinatorics
We say a positive integer $n$ is $k$-special if none of its figures is zero and, for any permutation the figures of $n$, the resulting number is multiple of $k$. Let $m\ge 2$ be a positive integer. [list] [*] Find the number of $4$-special numbers with $m$ figures. [*] Find the number of $3$-special numbers with $m$ figures. [/list]

2009 Middle European Mathematical Olympiad, 7
Tags: induction , combinatorics
The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.

2007 Romania Team Selection Test, 4
Tags: euler , inequalities , circumcircle , geometric transformation , homothety , cyclic quadrilateral , geometry
The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.

2019-IMOC, A1
Tags: algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $$f(xy+f(x))=f(xf(y))+x$$

2023 Moldova Team Selection Test, 2
Tags: number theory
Let $a,b,c$ be distinct positive integers and let $r,s,t$ be positive integers such that: $ab+1=r^2,ac+1=s^2,bc+1=t^2$ Prove that it is not possible that all three fractions$ \frac{rt}{s}, \frac{rs}{t}, \frac{st}{r}$ are integers.

2000 IMO Shortlist, 7
Tags: geometry , pythagorean theorem
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?

2024 All-Russian Olympiad Regional Round, 9.10
Tags: number theory
Does there exist a positive integer $n>10^{100}$, such that $n^2$ and $(n+1)^2$ satisfy the following property: every digit occurs equal number of times in the decimal representations of each number?

2009 Today's Calculation Of Integral, 483
Tags: calculus , integration , limit , function , calculus computations , logarithm
Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

2006 AMC 12/AHSME, 1
Tags:
Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? $ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$

2013 Stanford Mathematics Tournament, 13
Tags:
A board has $2$, $4$, and $6$ written on it. A person repeatedly selects (not necessarily distinct) values for $x$, $y$, and $z$ from the board, and writes down $xyz+xy+yz+zx+x+y+z$ if and only if that number is not yet on the board and is also less than or equal to $2013$. This person repeats this process until no more numbers can be written. How many numbers will be written at the end of the process?

2024 IMC, 2
Tags: real analysis , calculus , logarithm
For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

2007 Princeton University Math Competition, 2
Tags:
Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's [i]vicious hobgoblin +1 of viciousness,[/i] Hugo rolls $25$ $20$-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers was as large as possible. How many $2$s did Hugo roll?

2014 Dutch IMO TST, 2
Tags: trigonometry , geometry , geometric transformation , reflection , projective geometry , trig identities , law of sines
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

2000 Singapore Team Selection Test, 1
Tags: algebra , functional equation , functional
Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

May Olympiad L1 - geometry, 2006.2
Tags: geometry , rectangle , area , pentagon , paper
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

2016 AMC 12/AHSME, 16
Tags:
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

2016 Romania National Olympiad, 3
Tags: number theory , prime factorization
Find all the positive integers $p$ with the property that the sum of the first $p$ positive integers is a four-digit positive integer whose decomposition into prime factors is of the form $2^m3^n(m + n)$, where $m, n \in N^*$.

1982 Bulgaria National Olympiad, Problem 3
Tags: geometry , 3d geometry , prism , circumcircle
In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

2024 Kosovo Team Selection Test, P3
Tags: functional equation , algebra
Find all functions $f:\mathbb R\to\mathbb R$ such that $$(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)$$for all $x, y\in\mathbb R$.

2014-2015 SDML (Middle School), 14
Tags:
Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles? $\text{(A) }\frac{1}{16}\qquad\text{(B) }\frac{1}{9}\qquad\text{(C) }\frac{1}{8}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{4}$

2003 Miklós Schweitzer, 10
Tags: probability , function
Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$). (translated by L. Erdős)

2023 Korea - Final Round, 6
Tags: algebra , inequalities
For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following. $$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$ ($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)

2021 BMT, 6
Tags: geometry
Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]