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.

AND:
OR:
NO:

Found problems: 85335

2011 AMC 12/AHSME, 5
Tags:
Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

LMT Speed Rounds, 23
Tags: number theory
Let $S$ be the set of all positive integers $n$ such that the sum of all factors of $n$, including $1$ and $n$, is $120$. Compute the sum of all numbers in $S$. [i]Proposed by Evin Liang[/i]

2000 Brazil Team Selection Test, Problem 3
Tags: geometry
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.

2009 AIME Problems, 3
Tags: number theory , relatively prime , probability
A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

2021 OMMock - Mexico National Olympiad Mock Exam, 5
Tags: combinatorics
Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that: [list] [*] No two numbers are equal. [*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them. [/list] What is the greatest value of $k$ for which this is possible?

2019 Dürer Math Competition (First Round), P2
Tags: combinatorics
a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true. b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.

2024 HMNT, 9
Tags: guts
Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$

2018 Purple Comet Problems, 2
Tags: geometry
A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$. [img]https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/2018%2Bpc%2Bhs2.png[/img]

2019 Harvard-MIT Mathematics Tournament, 10
Tags: geometry , hmmt
In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon $A_4A_3B_4B_3C_4C_3$. Find the area of the hexagon $A_5A_6B_5B_6C_5C_6$.

1994 AMC 12/AHSME, 19
Tags:
Label one disk "$1$", two disks "$2$", three disks "$3$"$, ...,$ fifty disks "$50$". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 51 \qquad\textbf{(C)}\ 415 \qquad\textbf{(D)}\ 451 \qquad\textbf{(E)}\ 501 $

2012 India Regional Mathematical Olympiad, 6
Tags: subset , set theory , combinatorics of set , combinatorics
Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.) Find the sum of [i]WSUMs[/i] over all the subsets of S. (Assume that WSUM for the null set is $0$.)

1971 Canada National Olympiad, 9
Tags: quadratic , trigonometry , similar triangles , geometry
Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

1984 Polish MO Finals, 5
Tags: circles , hexagon , disks , combinatorial geometry , geometry
A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

2019 Balkan MO Shortlist, G3
Tags: tangent , circumcircle , geometry
Let $ABC$ be a scalene and acute triangle with circumcenter $O$. Let $\omega$ be the circle with center $A$, tangent to $BC$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $FG \perp AO$, $\angle BFD = \angle DGC$ and the couples of points $(B,F)$ and $(C,G)$ are in different halfplanes with respect to the line $AD$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $ABC$.

2015 Grand Duchy of Lithuania, 2
Tags: perpendicular , circles , geometry
Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.

2011 China Team Selection Test, 3
Tags: inequalities
Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

1999 Polish MO Finals, 2
Tags: combinatorics , pigeonhole principle
Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$

2000 AMC 8, 7
Tags:
What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$? $\text{(A)}\ -336 \qquad \text{(B)}\ -280 \qquad \text{(C)}\ -210 \qquad \text{(D)}\ -192 \qquad \text{(E)}\ 0$

2014 Germany Team Selection Test, 1
Tags: algorithm , additive combinatorics , combinatorics
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

2011 Akdeniz University MO, 3
Tags: inequality , inequalities
For all $x \geq 2$, $y \geq 2$ real numbers, prove that $$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$

2020 CMIMC Team, 7
Tags: team
Points $P$ and $Q$ lie on a circle $\omega$. The tangents to $\omega$ at $P$ and $Q$ intersect at point $T$, and point $R$ is chosen on $\omega$ so that $T$ and $R$ lie on opposite sides of $PQ$ and $\angle PQR = \angle PTQ$. Let $RT$ meet $\omega$ for the second time at point $S$. Given that $PQ = 12$ and $TR = 28$, determine $PS$.

2013 Online Math Open Problems, 25
Tags: modular arithmetic , calculus , integration
Positive integers $x,y,z \le 100$ satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute $10000x+100y+z$. [i]Evan Chen[/i]

2017 Finnish National High School Mathematics Comp, 3
Tags: digit , last digit , number theory
Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

2023-IMOC, N4
Tags: number theory
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $af(a)^3+2abf(a)+bf(b)$ is a perfect square for all positive integers $a,b$.

2024 JHMT HS, 6
Tags: combinatorics
Compute the number of nonempty subsets $S$ of $\{ 1,2,3,4,5,6,7,8,9,10 \}$ such that the median of $S$ is an element of $S$.