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

2020 JBMO Shortlist, 1
Tags: number theory
Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.

1969 AMC 12/AHSME, 34
Tags: algebra , polynomial , quadratic
The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as: $\textbf{(A) }2^{100}-1\qquad \textbf{(B) }2^{100}(x-1)-(x-2)\qquad \textbf{(C) }2^{100}(x-3)\qquad$ $\textbf{(D) }x(2^{100}-1)+2(2^{99}-1)\qquad \textbf{(E) }2^{100}(x+1)-(x+2)$

2009 USAMTS Problems, 2
Tags:
Alice has three daughters, each of whom has two daughters, each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)

2022 Durer Math Competition (First Round), 2
Tags: geometry , circumcircle
In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP \cdot OQ$.

2011 QEDMO 9th, 1
Tags: perfect square , number theory
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.

2020 USAMTS Problems, 2:
Tags:
[b]2/1/32.[/b] Is it possible to fill in a $2020$ x $2020$ grid with the integers from $1$ to $4,080,400$ so that the sum of each row is $1$ greater than the previous row?

2023 Iranian Geometry Olympiad, 5
Tags: geometry
There are $n$ points in the plane such that at least $99\%$ of quadrilaterals with vertices from these points are convex. Can we find a convex polygon in the plane having at least $90\%$ of the points as vertices? [i]Proposed by Morteza Saghafian - Iran[/i]

2016 Gulf Math Olympiad, 3
Tags: geometry
Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side $BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts $AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$. (i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$. (ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre of $ABC$ lies on the line $PQ$.

2015 Brazil Team Selection Test, 4
Tags: geometry
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

2019 BMT Spring, 6
Tags: geometry
How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.

2024 China Team Selection Test, 18
Tags: combinatorics , inequalities
Let $m,n\in\mathbb Z_{\ge 0},$ $a_0,a_1,\ldots ,a_m,b_0,b_1,\ldots ,b_n\in\mathbb R_{\ge 0}$ For any integer $0\le k\le m+n,$ define $c_k:=\max_{i+j=k}a_ib_j.$ Proof $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$ [i]Created by Yinghua Ai[/i]

2016 PUMaC Geometry A, 5
Tags: geometry
Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

2024 Indonesia TST, C
Tags: subset , combinatorics
Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

2017 Dutch IMO TST, 4
Tags: functional equation , algebra
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

2010 Putnam, A3
Tags: function , calculus , derivative , complex analysis , probability , vector
Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

2015 Purple Comet Problems, 19
Tags: geometry , similar triangles
Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z. For diagram to http://www.purplecomet.org/welcome/practice

1994 IMO Shortlist, 2
Tags: midpoint , geometry , quadrilateral
$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

1995 Korea National Olympiad, Problem 3
Tags: geometry , inradius , equilateral triangle , maximum
Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

2010 Saudi Arabia Pre-TST, 4.2
Tags: triangle inequality , geometry , area
Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.

2005 District Olympiad, 4
Tags: algebra
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$. a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$; b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.

2023 Baltic Way, 17
Tags: number theory
Find all pairs of positive integers $(a, b)$, such that $S(a^{b+1})=a^b$, where $S(m)$ denotes the digit sum of $m$.

2017 Princeton University Math Competition, A7
Tags:
If $N$ is the number of ways to place $16$ [i]jumping [/i]rooks on an $8 \times 8$ chessboard such that each rook attacks exactly two other rooks, find the remainder when $N$ is divided by $1000$. (A jumping rook is said to [i]attack [/i]a square if the square is in the same row or in the same column as the rook.)

2021 Romania EGMO TST, P4
Tags: combinatorics , geometry
Consider a coordinate system in the plane, with the origin $O$. We call a lattice point $A{}$ [i]hidden[/i] if the open segment $OA$ contains at least one lattice point. Prove that for any positive integer $n$ there exists a square of side-length $n$ such that any lattice point lying in its interior or on its boundary is hidden.

2017 ELMO Shortlist, 2
Tags: geometry
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$. [i]Proposed by Vincent Huang

1990 IMO Longlists, 58
Tags: perfect square , complex number , algebra , convex polygon
Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.