Found problems: 85335
2013 Stanford Mathematics Tournament, 2
Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running low on ingredients. Each pizza must have 1 lb of dough, $\frac14$ lb of cheese, $\frac16$ lb of sauce, and $\frac13$ lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that JImmy can make?
2019 Nigeria Senior MO Round 2, 5
Let $a$, $b$, and $c$ be real numbers such that $abc=1$. prove that
$\frac{1+a+ab}{1+b+ab}$ +$\frac{1+b+bc}{1+c+bc}$ + $\frac{1+c+ac}{1+a+ac}$ $>=3$
2020 Latvia Baltic Way TST, 13
It is given that $n$ and $\sqrt{12n^2+1}$ are both positive integers. Prove that:
$$ \sqrt{ \frac{\sqrt{12n^2+1}+1}{2}} $$
is also positive integer.
2022 CCA Math Bonanza, TB2
Determine the last three digits of $374^{2022}.$
[i]2022 CCA Math Bonanza Tiebreaker Round #2[/i]
1999 All-Russian Olympiad Regional Round, 10.6
Triangle $ABC$ has an inscribed circle tangent to sides $AB$, $AC$ and $BC$ at points $C_1$, $B_1$ and $A_1 $ respectively. Let $K$ be a point on the circle diametrically opposite to point $C_1$, $D$ be the intersection point of lines $B_1C_1$ and $A_1K$. Prove that $CD = CB_1$.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2011 China Girls Math Olympiad, 8
The $A$-excircle $(O)$ of $\triangle ABC$ touches $BC$ at $M$. The points $D,E$ lie on the sides $AB,AC$ respectively such that $DE\parallel BC$. The incircle $(O_1)$ of $\triangle ADE$ touches $DE$ at $N$. If $BO_1\cap DO=F$ and $CO_1\cap EO=G$, prove that the midpoint of $FG$ lies on $MN$.
2024 Princeton University Math Competition, A2 / B4
Arnie draws $20$ real numbers independently and uniformly at random from the interval $[0, 1].$ Given that the largest number that Arnie draws equals $\tfrac{19}{20},$ the expected value of the average of the $20$ numbers can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$
2020 Dürer Math Competition (First Round), P5
Let $p$ be prime and $ k > 1$ be a divisor of $p-1$. Show that if a polynomial of degree $k$ with integer coefficients attains every possible value modulo $ p$ that is $(0,1,\dots, p-1)$ at integer inputs then its leading coefficient must be divisible by $p$.
[hide=Note]Note: the leading coefficient of a polynomial of degree d is the coefficient of the $x_d$ term.[/hide]
2014 Sharygin Geometry Olympiad, 4
Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?
(A. Garkavyj, A. Sokolov )
2006 Iran MO (3rd Round), 1
For positive numbers $x_{1},x_{2},\dots,x_{s}$, we know that $\prod_{i=1}^{s}x_{k}=1$. Prove that for each $m\geq n$ \[\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}\]
1984 IMO Longlists, 42
Triangle $ABC$ is given for which $BC = AC + \frac{1}{2}AB$. The point $P$ divides $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2\angle CPA$.
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
2021 Alibaba Global Math Competition, 13
Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$.
(1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$.
(2) $M_n$ is Lie Group if and only if $n=2$.
2015 IMO Shortlist, A1
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
2024 Singapore Junior Maths Olympiad, Q5
Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$
Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]
2012 Puerto Rico Team Selection Test, 2
A cone is constructed with a semicircular piece of paper, with radius 10. Find the
height of the cone.
1992 Tournament Of Towns, (332) 4
$10$ numbers are placed on a circle. Their sum is equal to $100$. A sum of any three neighbouring numbers is no less than $29$. Find the minimal number $A$ such that for any such set of 10 numbers none of them is greater than $A$. Prove that this value for $A$ is really minimal.
(A. Tolpygo, Kiev)
2004 Estonia National Olympiad, 5
Real numbers $a, b$ and $c$ satisfy $$\begin{cases} a^2 + b^2 + c^2 = 1 \\ a^3 + b^3 + c^3 = 1. \end{cases}$$ Find $a + b + c$.
Denmark (Mohr) - geometry, 2010.5
An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts.
[img]https://1.bp.blogspot.com/-hQQV-Of96Ls/XzXCZjCledI/AAAAAAAAMV0/SwXa4mtEEm04onYbFGZiTc5NSpkoyvJLwCLcBGAsYHQ/s0/2010%2BMohr%2Bp5.png[/img]
2017 AIME Problems, 8
Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2021 SEEMOUS, Problem 4
For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.
2005 Romania National Olympiad, 1
We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube.
[i]Dinu Serbanescu[/i]
2019 Iran Team Selection Test, 2
Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers?
[i]Proposed by Morteza Saghafian [/i]
2008 ISI B.Stat Entrance Exam, 8
In how many ways can you divide the set of eight numbers $\{2,3,\cdots,9\}$ into $4$ pairs such that no pair of numbers has $\text{gcd}$ equal to $2$?