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

2014 Contests, 1
Tags: number theory
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]

2009 Tuymaada Olympiad, 1
Tags: modular arithmetic , algebra
A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

1996 Canadian Open Math Challenge, 3
Tags: geometry , inradius , circumcircle
The vertices of a right-angled triangle are on a circle of radius $R$ and the sides of the triangle are tangent to another circle of radius $r$ (this is the circle that is inside triangle). If the lengths of the sides about the right angles are 16 and 30, determine the value of $R+r$.

2021 Yasinsky Geometry Olympiad, 6
Tags: geometry , collinear , cyclic quadrilateral
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line. (Olena Artemchuk)

2021 MOAA, 14
Tags: team
Evaluate \[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\] [i]Proposed by Nathan Xiong[/i]

1988 Polish MO Finals, 3
Tags: symmetry , parallelogram , geometry
$W$ is a polygon which has a center of symmetry $S$ such that if $P$ belongs to $W$, then so does $P'$, where $S$ is the midpoint of $PP'$. Show that there is a parallelogram $V$ containing $W$ such that the midpoint of each side of $V$ lies on the border of $W$.

2011 AIME Problems, 4
Tags: geometry , geometric transformation , homothety , trigonometry , angle bisector , trig identities , law of sines
In triangle $ABC$, $AB=125,AC=117$, and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.

1999 All-Russian Olympiad Regional Round, 11.4
Tags: geometry , 3d geometry , sphere
A polyhedron is circumscribed around a sphere. Let's call its face [i]large [/i] if the projection of the sphere onto the plane of the face falls entirely within the face. Prove that there are no more than 6 large faces.

2007 Czech-Polish-Slovak Match, 2
Tags: number theory
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m.$

1967 All Soviet Union Mathematical Olympiad, 091
Tags: combinatorics , chessboard
"KING-THE SUICIDER" Given a chess-board $1000\times 1000$, $499$ white castles and a black king. Prove that it does not matter neither the initial situation nor the way white plays, but the king can always enter under the check in a finite number of moves.

1942 Putnam, B2
Tags: parabola , conic
For the family of parabolas $$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$ (i) find the locus of vertices, (ii) find the envelope, (iii) sketch the envelope and two typical curves of the family.

III Soros Olympiad 1996 - 97 (Russia), 10.3
Tags: analytic geometry
Let's consider the graph of a square trinomial having roots $1$ and $4$. Let's draw two tangents to it from point $O$ (the origin of coordinates), touching it at points $A$ and $B$. What values can the cosine of angle $\angle AOB$ take?

2014 VTRMC, Problem 3
Tags: number theory
Find the least positive integer $n$ such that $2^{2014}$ divides $19^n-1$.

2017 Balkan MO Shortlist, G7
Tags: geometry , concurrency
Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.

2011 Romania Team Selection Test, 4
Tags: inequalities , ratio , trigonometry , geometry
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

2016 Korea Summer Program Practice Test, 7
Tags: algebra , parallelogram , geometry
A infinite sequence $\{ a_n \}_{n \ge 0}$ of real numbers satisfy $a_n \ge n^2$. Suppose that for each $i, j \ge 0$ there exist $k, l$ with $(i,j) \neq (k,l)$, $l - k = j - i$, and $a_l - a_k = a_j - a_i$. Prove that $a_n \ge (n + 2016)^2$ for some $n$.

2024 LMT Fall, 5
Tags: speed
Find the area of the quadrilateral with vertices at $(0,0), (2,0), (20,24), (0,2)$ in that order.

2014 Vietnam National Olympiad, 3
Tags: geometry , circumcircle , combinatorics
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices. a) Find all possible values of pair $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

2018 Federal Competition For Advanced Students, P2, 2
Tags: geometric inequality , geometry , cyclic quadrilateral
Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds. [i](Proposed by Karl Czakler)[/i]

1991 Polish MO Finals, 1
Tags: geometry
Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that: (i) the volume of $T_1$ is greater than that of $T_2$; (ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$.

2001 Moldova National Olympiad, Problem 7
Tags: sequence , algebra
Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.

2014 Turkey MO (2nd round), 1
Tags: absolute value , combinatorics
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?

1980 IMO Longlists, 15
Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1954 AMC 12/AHSME, 36
Tags: ratio
A boat has a speed of $ 15$ mph in still water. In a stream that has a current of $ 5$ mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is: $ \textbf{(A)}\ \frac{5}{4} \qquad \textbf{(B)}\ \frac{1}{1} \qquad \textbf{(C)}\ \frac{8}{9} \qquad \textbf{(D)}\ \frac{7}{8} \qquad \textbf{(E)}\ \frac{9}{8}$

2010 IFYM, Sozopol, 8
Tags: geometry , trapezoid , equal segments , equal angles
In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.