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

Ukrainian From Tasks to Tasks - geometry, 2014.9

On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.

2020 Jozsef Wildt International Math Competition, W1

Consider the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1$$($a$ and $b > 0$) and the ellipse $E$ which is the intersection of the ellipsoid with the plane of equation$$mx + ny + pz = 0$$where the point $P = [m, n, p]$ is a random point from the unit sphere $(m^2 + n^2 + p^2 = 1)$. Consider the random variable $A_E$ the area of the ellipse $E$. If the point $P$ is chosen with uniform distribution with respect to the area on the unit sphere, what is the expectation of $A_E$ ?

2000 Switzerland Team Selection Test, 1

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

1952 AMC 12/AHSME, 19

Tags:
Angle $ B$ of triangle $ ABC$ is trisected by $ BD$ and $ BE$ which meet $ AC$ at $ D$ and $ E$ respectively. Then: $ \textbf{(A)}\ \frac {AD}{EC} \equal{} \frac {AE}{DC} \qquad\textbf{(B)}\ \frac {AD}{EC} \equal{} \frac {AB}{BC} \qquad\textbf{(C)}\ \frac {AD}{EC} \equal{} \frac {BD}{BE}$ $ \textbf{(D)}\ \frac {AD}{EC} \equal{} \frac {AB\cdot BD}{BE\cdot BC} \qquad\textbf{(E)}\ \frac {AD}{EC} \equal{} \frac {AE\cdot BD}{DC\cdot BE}$

2004 IMO Shortlist, 1

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

1966 Swedish Mathematical Competition, 3

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

2012 CIIM, Problem 5

Tags: function
Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$ A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$. Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.

2022 Baltic Way, 6

Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

2015 Saudi Arabia BMO TST, 2

Given $2015$ subsets $A_1, A_2,...,A_{2015}$ of the set $\{1, 2,..., 1000\}$ such that $|A_i| \ge 2$ for every $i \ge 1$ and $|A_i \cap A_j| \ge 1$ for every $1 \le i < j \le 2015$. Prove that $k = 3$ is the smallest number of colors such that we can always color the elements of the set $\{1, 2,..., 1000\}$ by $k$ colors with the property that the subset $A_i$ has at least two elements of different colors for every $i \ge 1$. Lê Anh Vinh

2014 Postal Coaching, 5

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

2014 Chile National Olympiad, 5

Prove that if a quadrilateral $ABCD$ can be cut into a finite number of parallelograms, then $ABCD$ is a parallelogram.

2014 ASDAN Math Tournament, 16

Tags:
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.

2022 Bolivia IMO TST, P2

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2012 Czech-Polish-Slovak Junior Match, 3

Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

2013 BMT Spring, 18

Paul and his pet octahedron like to play games together. For this game, the octahedron randomly draws an arrow on each of its faces pointing to one of its three edges. Paul then randomly chooses a face and progresses from face to adjacent face, as determined by the arrows on each face, and he wins if he reaches every face of the octahedron. What is the probability that Paul wins?

PEN A Problems, 38

Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.

2016 Miklós Schweitzer, 6

Let $\Gamma(s)$ denote Euler's gamma function. Construct an even entire function $F(s)$ that does not vanish everywhere, for which the quotient $F(s)/\Gamma(s)$ is bounded on the right halfplane $\{\Re(s)>0\}$.

1996 AMC 8, 7

Tags:
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has $4$ goldfish at the same time that Gretel has $128$ goldfish, then in how many months from that time will they have the same number of goldfish? $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

2023 ITAMO, 2

Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows: -for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$, -he erases all integers on the board, -he writes on the board the numbers $a_1, a_2,\ldots , a_n$. For instance, if $n = 5$ and the numbers initially on the board are $0, 7, 2, 6, 2$, after the first move the numbers on the board will be $1, 3, 3, 3, 3$, after the second they will be $1, 1, 5, 5, 5$, and so on. (a) Show that, whatever $n$ and whatever the initial configuration, the numbers on the board will eventually not change any more. (b) As a function of $n$, determine the minimum integer $k$ such that, whatever the initial configuration, moves from the $k$-th onwards will not change the numbers written on the board.

2015 Thailand TSTST, 2

In any $\vartriangle ABC, \ell$ is any line through $C$ and points $P, Q$. If $BP, AQ$ are perpendicular to the line $\ell$ and $M$ is the midpoint of the line segment $AB$, then prove that $MP = MQ$

2023-24 IOQM India, 30

Tags:
Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$

2014 Contests, 1

Tags: algebra , function
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

2005 Sharygin Geometry Olympiad, 18

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$. a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point, b) where can this intersection point lie?

1971 Swedish Mathematical Competition, 4

Tags: algebra
Find \[ \frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771} \]

1968 All Soviet Union Mathematical Olympiad, 096

Tags: geometry
The circumference with the radius $100$ cm is drawn on the cross-lined paper with the side of the squares $1$ cm. It neither comes through the vertices of the squares, nor touches the lines. How many squares can it pass through?