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

MOAA Gunga Bowls, 2021.22
Tags:
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$. [i]Proposed by Andy Xu[/i]

2024 Indonesia TST, 2
Tags: geometry
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

1971 Putnam, A3
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The three vertices of a triangle of sides $a,b,$ and $c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R.$ (Lattice points are points in Euclidean plane with integral coordinates.)

VI Soros Olympiad 1999 - 2000 (Russia), 10.5
Tags: geometry , fixed , fixed point
Two different points $A$ and $B$ have been marked on the circle $\omega$. We consider all points $X$ of the circle $\omega$, different from $A$ and $B$. Let $Y$ be the middpoint of the chord $AX$ and $Z$ be the projection of point $A$ on the line $BX$. Prove that all straight lines $YZ$ pass through a certain fixed point that does not depend on the choice of point $X$.

1972 AMC 12/AHSME, 8
Tags: logarithm
If $|x-\log y|=x+\log y$ where $x$ and $\log y$ are real, then $\textbf{(A) }x=0\qquad\textbf{(B) }y=1\qquad\textbf{(C) }x=0\text{ and }y=1\qquad$ $\textbf{(D) }x(y-1)=0\qquad \textbf{(E) }\text{None of these}$

2011 Saint Petersburg Mathematical Olympiad, 5
Tags: number theory
Let $M(n)$ and $m(n)$ are maximal and minimal proper divisors of $n$ Natural number $n>1000$ is on the board. Every minute we replace our number with $n+M(n)-m(n)$. If we get prime, then process is stopped. Prove that after some moves we will get number, that is not divisible by $17$

1996 IMO, 1
Tags: rectangle , combinatorics , graph theory , modular arithmetic
We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?

2013 Kazakhstan National Olympiad, 3
Tags: induction , inequalities , algebra
Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$

2003 Korea - Final Round, 3
Tags: algebra
Show that the equation, $2x^4+2x^2y^2+y^4=z^2$, does not have integer solution when $x \neq 0$.

2015 Turkey Junior National Olympiad, 4
Tags: geometry , geometric transformation , reflection
Let $ABC$ be a triangle and $D$ be the midpoint of the segment $BC$. The circle that passes through $D$ and tangent to $AB$ at $B$, and the circle that passes through $D$ and tangent to $AC$ at $C$ intersect at $M\neq D$. Let $M'$ be the reflection of $M$ with respect to $BC$. Prove that $M'$ is on $AD$.

2012 Dutch IMO TST, 1
Tags: incenter , geometry
A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$. [i]S. Berlov[/i]

PEN O Problems, 17
Tags:
Find the maximum number of pairwise disjoint sets of the form \[S_{a,b}=\{n^{2}+an+b \; \vert \; n \in \mathbb{Z}\},\] with $a,b \in \mathbb{Z}$.

1958 Kurschak Competition, 3
Tags: equal area , geometry , hexagon
The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area

2021 Polish Junior MO First Round, 2
Tags: geometry , isosceles , altitude
A triangle $ABC$ is given with $AC = BC = 5$. The altitude of this triangle drawn from vertex $A$ has length $4$. Calculate the length of the altitude of $ABC$ drawn from vertex $C$.

1990 National High School Mathematics League, 9
Tags: inequalities
Let $n$ be a natural number. For all real numbers $x,y,z$, $(x^2+y^2+z^2)^2\geq n(x^4+y^4+z^4)$, then the minumum value of $n$ is________.

2017 China Northern MO, 4
Tags: graph theory , combinatorics
Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).

1950 AMC 12/AHSME, 7
Tags:
If the digit $1$ is placed after a two digit number whose tens' digit is $t$, and units' digit is $u$, the new number is: $\textbf{(A)}\ 10t+u+1 \qquad \textbf{(B)}\ 100t+10u+1 \qquad \textbf{(C)}\ 100t+10u+1\qquad \textbf{(D)}\ t+u+1 \qquad \textbf{(E)}\ \text{None of these answers}$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3
Tags: number theory
Find all pairs of prime natural numbers $(p, q)$ for which the value of the expression $\frac{p}{q}+\frac{p+1}{q+1}$ is an integer.

2015 Purple Comet Problems, 3
Tags:
The Fahrenheit temperature ($F$) is related to the Celsius temperature ($C$) by $F = \tfrac{9}{5} \cdot C + 32$. What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees?

2006 Petru Moroșan-Trident, 2
Tags: linear algebra , matrix
Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $ [b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal. [b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element. [i]Gheorghe Iurea[/i] [hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]

2009 IMS, 4
Tags: ratio , vector , algebra , polynomial , limit , induction , absolute value
In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

2022 Math Prize for Girls Problems, 15
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What is the smallest positive integer $m$ such that $15! \, m$ can be expressed in more than one way as a product of $16$ distinct positive integers, up to order?

Estonia Open Junior - geometry, 2011.2.3
Tags: geometry , hexagon , regular
Consider the diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_4$ and $A_6A_2$ of a convex hexagon $A_1A_2A_3A_4A_5A_6$. The hexagon whose vertices are the points of intersection of the diagonals is regular. Can we conclude that the hexagon $A_1A_2A_3A_4A_5A_6$ is also regular?

2023 Federal Competition For Advanced Students, P2, 3
Tags: combinatorics , combinatorial geometry , geometry
Alice and Bob play a game, in which they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shares at least one point - except for its starting point - with one of the previously drawn segments, one has lost. a) Show that both Alice and Bob could force the game to end, if they don’t care who wins. b) Is there a winning strategy for one of them?

2022 Thailand Online MO, 8
Tags: geometry , circumcircle
Let $ABCD$ be a convex quadrilateral with $AD = BC$, $\angle BAC+\angle DCA = 180^{\circ}$, and $\angle BAC \neq 90^{\circ}.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABC$ and $CAD$, respectively. Prove that one intersection point of the circumcircles of triangles $O_1BC$ and $O_2AD$ lies on $AC$.