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

2009 Belarus Team Selection Test, 2

Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ? I. Voronovich

2019 USMCA, 2

Tags:
Let $n \ge 2$ be an even integer. Find the maximum integer $k$ (in terms of $n$) such that $2^k$ divides $\binom{n}{m}$ for some $0 \le m \le n$.

1969 Canada National Olympiad, 4

Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]

1965 Miklós Schweitzer, 7

Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.

2021 Israel TST, 3

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

2024 Thailand TST, 1

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

MathLinks Contest 3rd, 2

The sequence $\{x_n\}_{n\ge1}$ is defined by $x_1 = 7$, $x_{n+1} = 2x^2_n - 1$, for all positive integers $n$. Prove that for all positive integers $n$ the number $x_n$ cannot be divisible by $2003$.

2022 South East Mathematical Olympiad, 2

In acute triangle ABC AB>AC. H is the orthocenter. M is midpoint of BC and AD is the symmedian line. Prove that if $\angle ADH= \angle MAH$, EF bisects segment AD. [img]https://s2.loli.net/2022/08/02/t9xzTV8IEv1qQRm.jpg[/img]

1967 IMO Shortlist, 6

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

2016 Baltic Way, 4

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n | a + b + c + d$ and $n | a^2 + b^2 + c^2 + d^2. $ Show that $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$

2011 Israel National Olympiad, 3

In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called [b]uniform[/b] if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?

1941 Moscow Mathematical Olympiad, 084

a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$. b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.

2016 239 Open Mathematical Olympiad, 8

Given a natural number $k>1$. Find the smallest number $\alpha$ satisfying the following condition. Suppose that the table $(2k + 1) \times (2k + 1)$ is filled with real numbers not exceeding $1$ in absolute value, and the sums of the numbers in all lines are equal to zero. Then you can rearrange the numbers so that each number remains in its row and all the sums over the columns will be at most $\alpha$.

2019 LIMIT Category C, Problem 5

Tags: probability
Suppose that $X\sim\operatorname{Uniform}(0,1)$ and $Y\sim\operatorname{Bernoulli}\left(\frac14\right)$, independently of each other. Let $Z=X+Y$. Then which of the following is true? $\textbf{(A)}~\text{The distribution of }Z\text{ is symmetric about }1$ $\textbf{(B)}~Z\text{ has a probability density function}$ $\textbf{(C)}~E(Z)=\frac54$ $\textbf{(D)}~P(Z\le1)=\frac14$

2002 AIME Problems, 9

Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,10\}.$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}.$ (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000.$

2008 Junior Balkan Team Selection Tests - Romania, 4

Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2\plus{}AN^2 \equal{}BM^2 \plus{} CN^2 \textrm{ and } \angle MDN \equal{} \angle BAC.\] Prove that $ \angle BAC \equal{} 90^\circ$.

2025 ISI Entrance UGB, 8

Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$. Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.

1992 Polish MO Finals, 3

Show that for real numbers $x_1, x_2, ... , x_n$ we have: \[ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0 \] When do we have equality?

2006 Mathematics for Its Sake, 1

Determine the number of polynomials of degree $ 3 $ that are irreducible over the field of integers modulo a prime.

2023 HMNT, 16

Tags:
Compute the number of tuples $(a_0, a_1, a_2, a_3, a_4, a_5)$ of (not necessarily positive) integers such that $a_i \le i$ for all $0 \le i \le 5$ and $$a_0+a_1+a_2+a_3+a_4+a_5=6.$$

1969 IMO Longlists, 11

$(BUL 5)$ Let $Z$ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of $Z.$ Let us call such a pair of points unjoinable. Prove that for each real $r > 0$ there exists an unjoinable pair of points separated by distance $r.$

1961 Putnam, A7

Tags: topology , subset
Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$

1999 Hungary-Israel Binational, 2

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

2020 Online Math Open Problems, 11

Tags:
Let $ABC$ be a triangle such that $AB = 5$, $AC = 8$, and $\angle BAC = 60^{\circ}$. Let $P$ be a point inside the triangle such that $\angle APB = \angle BPC = \angle CPA$. Lines $BP$ and $AC$ intersect at $E$, and lines $CP$ and $AB$ intersect at $F$. The circumcircles of triangles $BPF$ and $CPE$ intersect at points $P$ and $Q \neq P$. Then $QE + QF=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$. [i]Proposed by Ankan Bhattacharya[/i]

2016 NIMO Summer Contest, 13

Tags: geometry
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$. [i]Proposed by Michael Tang[/i]