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

2013 Iran Team Selection Test, 12
Tags: geometry , incenter , circumcircle , quadratic , geometric transformation , homothety , trigonometry
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.

2008 National Olympiad First Round, 21
Tags: angle bisector , geometry
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P\in [AC]$, $Q\in [BC]$, $R\in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$, $M\in [AB]$, and $L\in [AC]$. What is $|AB|+|AC|$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16 $

2012 AMC 10, 19
Tags: algebra , linear equation
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break? $ \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 42 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 60 $

2017 Caucasus Mathematical Olympiad, 2
Tags: combinatorics
On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

2010 Federal Competition For Advanced Students, P2, 3
Tags: hexagon , combinatorics , tangent , combinatorial geometry
On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

2002 Iran Team Selection Test, 4
Tags: circumcircle , perimeter , geometry
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.

1992 Canada National Olympiad, 3
Tags: geometry , analytic geometry , trapezoid , similar triangles
In the diagram, $ ABCD$ is a square, with $ U$ and $ V$ interior points of the sides $ AB$ and $ CD$ respectively. Determine all the possible ways of selecting $ U$ and $ V$ so as to maximize the area of the quadrilateral $ PUQV$. [img]http://i250.photobucket.com/albums/gg265/geometry101/CMO1992Number3.jpg[/img]

2023 Federal Competition For Advanced Students, P1, 4
Tags: number theory
Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$

2012 Belarus Team Selection Test, 1
Tags: geometry , ratio , orthocenter , incenter , centroid
For any point $X$ inside an acute-angled triangle $ABC$ we define $$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$ where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ . (D. Bazylev)

2015 Poland - Second Round, 1
Tags: parallelogram , geometry
Points $E, F, G$ lie, and on the sides $BC, CA, AB$, respectively of a triangle $ABC$, with $2AG=GB, 2BE=EC$ and $2CF=FA$. Points $P$ and $Q$ lie on segments $EG$ and $FG$, respectively such that $2EP = PG$ and $2GQ=QF$. Prove that the quadrilateral $AGPQ$ is a parallelogram.

1994 IMO, 6
Tags: modular arithmetic , number theory , prime number , combinatorial number theory , combinatorics
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

2002 Iran MO (3rd Round), 24
Tags: geometry , incenter , circumcircle , geometric transformation , homothety , ratio , trapezoid
$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

2022 Canada National Olympiad, 3
Tags: combinatorics
Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.

2021 MOAA, 16
Tags: team
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

1976 Yugoslav Team Selection Test, Problem 3
Tags: inequalities
Find the minimum and maximum values of the function $$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.

2022 Vietnam TST, 2
Tags: polyhedron , combinatorics
Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number $26; 4$ and $2022$ (each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.

2016 Switzerland Team Selection Test, Problem 7
Tags: number theory , divisor
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$

1997 Romania National Olympiad, 2
Tags: number theory
Let $a \ne 0$ be a natural number. Prove that $a$ is a perfect square if and only if for every $b \in N^*$ there exists $c \in N^*$ such that $a + bc$ is a perfect square.

2008 India Regional Mathematical Olympiad, 2
Tags: number theory , least common multiple , inequalities , algorithm
Prove that there exist two infinite sequences $ \{a_n\}_{n\ge 1}$ and $ \{b_n\}_{n\ge 1}$ of positive integers such that the following conditions hold simultaneously: $ (i)$ $ 0 < a_1 < a_2 < a_3 < \cdots$; $ (ii)$ $ a_n < b_n < a_n^2$, for all $ n\ge 1$; $ (iii)$ $ a_n \minus{} 1$ divides $ b_n \minus{} 1$, for all $ n\ge 1$ $ (iv)$ $ a_n^2 \minus{} 1$ divides $ b_n^2 \minus{} 1$, for all $ n\ge 1$ [19 points out of 100 for the 6 problems]

MathLinks Contest 6th, 3.1
Tags: number theory
For each positive integer $n$ let $\tau (n)$ be the sum of divisors of $n$. Find all positive integers $k$ for which $\tau (kn - 1) \equiv 0$ (mod $k$) for all positive integers $n$.

2023 BMT, 6
Tags: number theory
Let $N$ be the number of positive integers $x$ less than $210 \cdot 2023$ such that $$ lcm(gcd(x, 1734), gcd(x + 17, x + 1732))$$ divides $2023$. Compute the sum of the prime factors of $N$ with multiplicity. (For example, if $S = 75 = 3^1 \cdot 5^2$, then the answer is $1\cdot 3 + 2 \cdot 5 = 13$).

2007 ITest, 33
Tags:
How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?

2008 CentroAmerican, 2
Tags: parallelogram , rectangle , geometry
Let $ ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $ O$ such that $ AC$ is a diameter. Pararellograms $ DAOE$ and $ BCOF$ are constructed. Show that if $ E$ and $ F$ lie on the circumference then $ ABCD$ is a rectangle.

2002 USAMTS Problems, 3
Tags:
Determine, with proof, the rational number $\dfrac{m}{n}$ that equals \[\tfrac{1}{1\sqrt2+2\sqrt1}+\tfrac{1}{2\sqrt3+3\sqrt2}+\tfrac{1}{3\sqrt4+4\sqrt3}+\ldots+\tfrac{1}{4012008\sqrt{4012009}+4012009\sqrt{4012008}}\]

1985 Iran MO (2nd round), 7
Tags: inequalities
Let $a,b$ and $c$ be real numbers with $b,c >0.$ Prove that if $ a<b \ ( a>b),$ then \[\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab) \] And then prove that $\frac{a+c}{b+c}$ is between $1$ and $\frac ab.$