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

1977 Putnam, A4
Tags:
For $0<x<1,$ express $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$ as a rational function of $x.$

2023 India National Olympiad, 2
Tags: polynomial , algebra , inequalities
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

2012 All-Russian Olympiad, 3
Tags: geometry , parallelogram , circumcircle , trapezoid , analytic geometry , graphing lines , slope
Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.

1993 Tournament Of Towns, (393) 1
Tags: geometry , geometric inequality
Two tangents $CA$ and $CB$ are drawn to a circle ($A$ and $B$ being the tangent points). Consider a “triangle” bounded by an arc $AB$ (the smaller one) and segments $CA$ and $CB$. Prove that the length of any segment inside the triangle is not greater than the length of $CA = CB$. (Folklore)

2024 Bulgarian Winter Tournament, 10.2
Tags: number theory
Find all positive integers $k$ for which there exist positive integers $x, y$, such that $\frac{x^ky}{x^2+y^2}$ is a prime.

1993 All-Russian Olympiad, 4
Tags: quadratic , geometry , geometric transformation
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?

1996 China Team Selection Test, 1
Tags: geometry , trapezoid
Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.

1998 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , analytic geometry , graphing lines , slope , derivative
Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

1999 Italy TST, 4
Tags: floor function , combinatorics
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that i) $|A_i|=3$ for each $i=1,\ldots ,m$. ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$. Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.

2008 Princeton University Math Competition, A10/B10
Tags: number theory
What is the smallest number $n$ such that you can choose $n$ distinct odd integers $a_1, a_2,..., a_n$, none of them $1$, with $\frac{1}{a_1}+ \frac{1}{a_2}+ ...+ \frac{1}{a_n}= 1$?

1986 IMO Shortlist, 9
Tags: graph theory , combinatorics , coloring , ramsey theory , extremal combinatorics
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

2020 CCA Math Bonanza, I13
Tags: number theory , arithmetic sequence
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$. [i]2020 CCA Math Bonanza Individual Round #13[/i]

2015 Saudi Arabia GMO TST, 3
Tags: orthocenter , circumcenter , perpendicular , geometry
Let $BD$ and $CE$ be altitudes of an arbitrary scalene triangle $ABC$ with orthocenter $H$ and circumcenter $O$. Let $M$ and $N$ be the midpoints of sides $AB$, respectively $AC$, and $P$ the intersection point of lines $MN$ and $DE$. Prove that lines $AP$ and $OH$ are perpendicular. Liana Topan

MOAA Gunga Bowls, 2023.17
Tags:
Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

2018 Hanoi Open Mathematics Competitions, 12
Tags: geometry , rectangle
Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$ BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$. [img]https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png[/img]

2004 Romania Team Selection Test, 16
Tags: inequalities , geometry , circumcircle , geometric transformation , homothety , parallelogram , complex number
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

2020 AMC 8 -, 15
Tags: percent
Suppose $15\%$ of $x$ equals $20\%$ of $y$. What percentage of $x$ is $y$? $\textbf{(A)}\ 5~~\qquad\textbf{(B)}\ 35~~\qquad~~\textbf{(C)}\ 75\qquad~~\textbf{(D)}\ 133\frac13\qquad~~ \textbf{(E)}\ 300$

Durer Math Competition CD Finals - geometry, 2017.D4
Tags: geometry , angle
The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$; a) Find the angle $\angle DPB$? b) Prove that $P$ lies on the diagonal $AC$.

2019 ELMO Shortlist, N5
Tags: number theory
Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

2016 AIME Problems, 7
Tags:
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB}\parallel \overline{EF}$. The area of $ABCD$ is $2016$, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest possible integer values of the area of $IJKL$.

2009 Moldova Team Selection Test, 3
Tags: geometry , geometric transformation , reflection , circumcircle , trigonometry , symmetry , incenter
[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]

2005 Romania Team Selection Test, 3
Tags: modular arithmetic , floor function , quadratic , induction , functional equation , number theory , algebra
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that \[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \] [i]Călin Popescu[/i]

2015 Taiwan TST Round 2, 2
Tags: function , algebra , functional equation
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

1993 Vietnam Team Selection Test, 1
Tags: geometry , incenter , circumcircle , inequalities , euler , ratio , angle bisector
Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?

2014 Contests, 3
Tags:
For how many integers $n$, there are four distinct real numbers satisfying the equation $ |x^2-4x-7|=n$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 5 $