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

2011 Iran MO (3rd Round), 2
Tags: polynomial , algebra
[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real. [b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real? [i]proposed by Mohammad Mansouri[/i]

2018 CHMMC (Fall), 3
Tags: number theory
Let $p$ be the third-smallest prime number greater than $5$ such that: $\bullet$ $2p + 1$ is prime, and $\bullet$ $5^p \not\equiv 1$ (mod $2p + 1$). Find $p$.

2007 Harvard-MIT Mathematics Tournament, 4
Tags: geometry
Circle $\omega$ has radius $5$ and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Compute $AB+AC$ given that $BC=7$.

2008 Harvard-MIT Mathematics Tournament, 12
Tags: ratio , ellipse , 3d geometry , sphere , conic , geometric series , geometry
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

PEN N Problems, 13
Tags: arithmetic sequence , more sequences , polynomial , algebra
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

2013 Sharygin Geometry Olympiad, 2
Tags: circumcircle , circles , geometry
Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

Mathley 2014-15, 3
Tags: projection , game strategy , circumcircle , orthocenter
A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$. Do Thanh Son, High School of Natural Sciences, National University, Hanoi

2009 Estonia Team Selection Test, 4
Tags: geometric inequalities , geometry , ratio , inequalities
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

2016 Middle European Mathematical Olympiad, 3
Tags: circumcenter , perpendicular lines , geometry
Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$. Prove that $\angle QCB = \angle PCO$.

2012 Saint Petersburg Mathematical Olympiad, 7
Tags: combinatorics , number theory
We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$). For what minimal $n$ it is always possible?

2017 Online Math Open Problems, 2
Tags:
The numbers $a,b,c,d$ are $1,2,2,3$ in some order. What is the greatest possible value of $a^{b^{c^d}}$? [i]Proposed by Yannick Yao and James Lin[/i]

2006 AMC 8, 1
Tags:
Mindy made three purchases for $ \$1.98, \$5.04$ and $ \$9.89$. What was her total, to the nearest dollar? $ \textbf{(A)}\ \$10 \qquad \textbf{(B)}\ \$15 \qquad \textbf{(C)}\ \$16 \qquad \textbf{(D)}\ \$17 \qquad \textbf{(E)}\ \$18$

2013 USA TSTST, 4
Tags: geometry , circumcircle , geometric transformation
Circle $\omega$, centered at $X$, is internally tangent to circle $\Omega$, centered at $Y$, at $T$. Let $P$ and $S$ be variable points on $\Omega$ and $\omega$, respectively, such that line $PS$ is tangent to $\omega$ (at $S$). Determine the locus of $O$ -- the circumcenter of triangle $PST$.

2008 AMC 8, 10
Tags:
The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? $\textbf{(A)}\ 32.5 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 33.5 \qquad \textbf{(D)}\ 34\qquad \textbf{(E)}\ 35$

2007 Spain Mathematical Olympiad, Problem 6
Tags: geometry , circles
Given a halfcircle of diameter $AB = 2R$, consider a chord $CD$ of length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$. Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.

JBMO Geometry Collection, 2005
Tags: radical axis , power of a point , geometry
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

2010 Tournament Of Towns, 3
Tags: inequalities , invariant
Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers. $(a)$ Find the minimal possible value of this sum. $(b)$ Find the maximal possible value of this sum.

2007 Sharygin Geometry Olympiad, 5
Tags: area , cut , polygon , geometry
A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to: a) five? b) four?

2003 District Olympiad, 4
Tags: real analysis , function
Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that: a)$f$ has at least one point of discontinuity; b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity; c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity. [i]Radu Mortici [/i]

2008 Harvard-MIT Mathematics Tournament, 8
Tags:
Trodgor the dragon is burning down a village consisting of $ 90$ cottages. At time $ t \equal{} 0$ an angry peasant arises from each cottage, and every $ 8$ minutes ($ 480$ seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor $ 5$ seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many [b]seconds[/b] does it take Trodgor to burn down the entire village?

1981 AMC 12/AHSME, 24
Tags: trigonometry
If $ \theta$ is a constant such that $ 0 < \theta < \pi$ and $ x \plus{} \frac{1}{x} \equal{} 2\cos{\theta}$. then for each positive integer $ n$, $ x^n \plus{} \frac{1}{x^n}$ equals $ \textbf{(A)}\ 2\cos{\theta}\qquad \textbf{(B)}\ 2^n\cos{\theta}\qquad \textbf{(C)}\ 2\cos^n{\theta}\qquad \textbf{(D)}\ 2\cos{n\theta}\qquad \textbf{(E)}\ 2^n\cos^n{\theta}$

1983 National High School Mathematics League, 8
Tags: circumcircle , inequalities , geometry
For any $\triangle ABC$, its girth is$l$, its circumradius is$R$, its inscribed radius is $r$.Which one is true? $\text{(A)}l>R+r\qquad\text{(B)}l\leq R+r\qquad\text{(C)}\frac{l}{6}<R+r<6l\qquad\text{(D)}$None above

1969 All Soviet Union Mathematical Olympiad, 118
Tags: algebra , inequalities
Given positive numbers $a,b,c,d$. Prove that the set of inequalities $$a+b<c+d$$ $$(a+b)(c+d)<ab+cd$$ $$(a+b)cd<ab(c+d)$$ contain at least one wrong.

1953 Miklós Schweitzer, 7
Tags: 3d geometry
[b]7.[/b] Consider four real numbers $t_{1},t_{2},t_{3},t_{4}$ such that each is less than the sum of the others. Show that there exists a tetrahedron whose faces have areas $t_{1},t_{2}, t_{3}$ and $t_{4},$ respectively. [b](G. 9)[/b]

2008 District Olympiad, 2
Tags: induction
Let $ S\equal{}\{1,2,\ldots,n\}$ be a set, where $ n\geq 6$ is an integer. Prove that $ S$ is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if $ n$ is a multiple of 3.