Found problems: 85335
2003 Flanders Junior Olympiad, 1
Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.
Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.
Who made it?
2020 Princeton University Math Competition, A3/B5
Find the sum (in base $10$) of the three greatest numbers less than $1000_{10}$ that are palindromes in both base $10$ and base $5$.
2007 All-Russian Olympiad, 5
Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero.
[i]N. Agakhanov[/i]
2011 District Olympiad, 2
Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there?
2025 USAJMO, 1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
[i]Note: [/i] A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
2010 Contests, 3
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/i]
2004 Purple Comet Problems, 1
How many different positive integers divide $10!$ ?
2010 Saudi Arabia IMO TST, 2
Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$
2013 Kyiv Mathematical Festival, 2
For every positive $a, b,c, d$ such that $a + c \le ac$ and $b + d \le bd$
prove that $\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4$
Kvant 2021, M2637
A table with three rows and 100 columns is given. Initially, in the left cell of each row there are $400\cdot 3^{100}$ chips. At one move, Petya marks some (but at least one) chips on the table, and then Vasya chooses one of the three rows. After that, all marked chips in the selected row are shifted to the right by a cell, and all marked chips in the other rows are removed from the table. Petya wins if one of the chips goes beyond the right edge of the table; Vasya wins if all the chips are removed. Who has a winning strategy?
[i]Proposed by P. Svyatokum, A. Khuzieva and D. Shabanov[/i]
2024 UMD Math Competition Part II, #4
Prove for every positive integer $n{:}$
\[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]
2022 CCA Math Bonanza, I13
Let triangle $A_1BC$ have sides $A_1B=5$, $A_1C=12$, and $BC=13$. For all natural numbers $i$, let $B_i$ be the foot of the altitude from $A_i$ to $BC$, let $A_{2i}$ be the foot of the altitude from $B_i$ to $A_1B$, and let $A_{2i+1}$ be the foot of the altitude from $B_i$ to $A_1C$.
\[ \sum_{i=1}^{7}A_iB_i = \frac{p}{q}\]
Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #13[/i]
2012 IFYM, Sozopol, 1
Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.
2002 Vietnam National Olympiad, 2
Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.
1949-56 Chisinau City MO, 17
Prove that if the roots of the equation $x^2 + px + q = 0$ are real, then for any real number $a$ the roots of the equation $$x^2 + px + q + (x + a) (2x + p) = 0$$ are also real.
2010 Stars Of Mathematics, 4
Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that
\[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \]
if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$.
(Dan Schwarz)
2025 Nordic, 3
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
2022 LMT Fall, 9
In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.
2010 Romania National Olympiad, 4
On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$.
[i]Nicolae Bourbacut[/i]
2021 MMATHS, 9
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$
[i]Proposed by Andrew Wu[/i]
2011 Uzbekistan National Olympiad, 1
Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$
2012 IMO Shortlist, G4
Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.
Novosibirsk Oral Geo Oly IX, 2017.1
Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya .
[img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]
2013 National Chemistry Olympiad, 55
What is the number of structural isomers with the molecular formula $\ce{C6H14}$?
$ \textbf{(A) }\text{Three} \qquad\textbf{(B) }\text{Four} \qquad\textbf{(C) }\text{Five} \qquad\textbf{(D) }\text{Six}\qquad$
2004 Iran MO (3rd Round), 22
Suppose that $ \mathcal F$ is a family of subsets of $ X$. $ A,B$ are two subsets of $ X$ s.t. each element of $ \mathcal{F}$ has non-empty intersection with $ A, B$. We know that no subset of $ X$ with $ n \minus{} 1$ elements has this property. Prove that there is a representation $ A,B$ in the form $ A \equal{} \{a_1,\dots,a_n\}$ and $ B \equal{} \{b_1,\dots,b_n\}$, such that for each $ 1\leq i\leq n$, there is an element of $ \mathcal F$ containing both $ a_i, b_i$.