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

2021 CMIMC, 1.6
Tags: algebra , number theory
Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$. [i]Proposed by Vijay Srinivasan[/i]

2016 AMC 8, 10
Tags:
Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $$2 * (5 * x)=1?$$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

2013 Czech-Polish-Slovak Junior Match, 1
Tags: diophantine equation , diophantine , number theory , radical
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2006 AMC 8, 13
Tags:
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? $ \textbf{(A)}\ 10: 00 \qquad \textbf{(B)}\ 10: 15 \qquad \textbf{(C)}\ 10: 30 \qquad \textbf{(D)}\ 11: 00 \qquad \textbf{(E)}\ 11: 30$

2015 AMC 12/AHSME, 16
Tags: geometry , 3d geometry , pyramid , symmetry , pythagorean theorem
A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? $\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$

2003 AMC 12-AHSME, 17
Tags: function , logarithm
If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$? $ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{3}{5} \qquad \textbf{(E)}\ 1$

1970 IMO Longlists, 50
Tags: geometry , perimeter , circumcircle , inequalities , area of a triangle
The area of a triangle is $S$ and the sum of the lengths of its sides is $L$. Prove that $36S \leq L^2\sqrt 3$ and give a necessary and sufficient condition for equality.

1998 Slovenia National Olympiad, Problem 1
Tags: number theory
Find all positive integers $n$ that are equal to the sum of digits of $n^2$.

2016 Indonesia TST, 1
Tags: number theory , abstract algebra , set
Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.

1987 AMC 8, 17
Tags:
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says: "Bret is next to Carl." "Abby is between Bret and Carl." However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2? $\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}$

Kyiv City MO Seniors 2003+ geometry, 2004.11.2
Tags: geometry , perpendicular bisector
Given a triangle $ABC$, in which $\angle B> 90^o$. Perpendicular bisector of the side $AB$ intersects the side $AC$ at the point $M$, and the perpendicular bisector of the side $AC$ intersects the extension of the side $AB$ beyond the vertex $B$ at point $N$. It is known that the segments $MN$ and $BC$ are equal and intersect at right angles. Find the values ​​of all angles of triangle $ABC$.

1982 Bundeswettbewerb Mathematik, 1
Tags: sum , number theory , divisor
Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

1981 National High School Mathematics League, 8
Tags: logarithm
In the logarithm table below, there are two mistakes. Correct them. \begin{tabular}{|c|c|} \hline % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $\lg0.021$&$2a+b+c-3$ \\ \hline $\lg0.27$&$6a-3b-2$\\ \hline $\lg1.5$&$3a-b+c$\\ \hline $\lg2.8$&$1-2a+2b-c$\\ \hline $\lg3$&$2a-b$\\ \hline $\lg5$&$a+c$\\ \hline $\lg6$&$1+a-b-c$\\ \hline $\lg7$&$2(a+c)$\\ \hline $\lg8$&$3-3a-3c$\\ \hline $\lg9$&$4a-2b$\\ \hline $\lg14$&$1-a+2b$\\ \hline \end{tabular}

2018 IMO Shortlist, A2
Tags: algebra , sequence , system of equations
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

2007 Puerto Rico Team Selection Test, 5
Tags: number theory , digit
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

2011 Belarus Team Selection Test, 4
Tags: algebra , system of equations
Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$) a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$ b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)? D. Bazylev

2018 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , digit , palindrome
Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values ​​of $N-M$.

2021 Winter Stars of Mathematics, 4
Tags: algebra , number theory
Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

2013 Tuymaada Olympiad, 1
Tags: combinatorics
$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy. [i]K. Kokhas[/i] [b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.

1998 National Olympiad First Round, 6
Tags: modular arithmetic
Find the number of primes$ p$, such that $ x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right)$ has no three distinct integer roots in $ \left[0,\left. p\right)\right.$. $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$

2020 LMT Fall, A23
Tags:
There are $5$ people left in a game of Among Us, $4$ of whom are crewmates and the last is the impostor. None of the crewmates know who the impostor is. The person with the most votes is ejected, unless there is a tie in which case no one is ejected. Each of the $5$ remaining players randomly votes for someone other than themselves. The probability the impostor is ejected can be expressed as $\frac{m}{n}$. Find $m+n$. [i]Proposed by Sammy Charney[/i]

2003 USA Team Selection Test, 2
Tags: ratio , geometry
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that \[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \] if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)

2005 Thailand Mathematical Olympiad, 20
Tags: max , algebra , inequalities
Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2007 Princeton University Math Competition, 1
Tags: modular arithmetic
Find the last three digits of \[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]

2023 Austrian MO National Competition, 3
Tags: combinatorics
Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.