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

2016 Kosovo Team Selection Test, 1
Tags: equation
Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

1950 Poland - Second Round, 5
Tags: geometry , construction , trisection , concentric
Given two concentric circles and a point $A$. Through point $A$, draw a secant such that its segment contained by the larger circle is divided by the smaller circle into three equal parts.

1985 Iran MO (2nd round), 4
Tags: group theory , abstract algebra , superior algebra
Let $G$ be a group and let $a$ be a constant member of it. Prove that \[G_a = \{x | \exists n \in \mathbb Z , x=a^n\}\] Is a subgroup of $G.$

2024 Taiwan TST Round 1, C
Tags: combinatorics
A $k$-set is a set with exactly $k$ elements. For a $6$-set $A$ and any collection $\mathcal{F}$ of $4$-sets, we say that $A$ is [i]$\mathcal{F}$-good[/i] if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy $$(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.$$ Find all $n \geq 6$ so that there exists a collection $\mathcal{F}$ of $4$-subsets of $\{1, 2, \ldots , n\}$ such that every $6$-set $A \subseteq \{1, 2, \ldots , n\}$ is $\mathcal{F}$-good. [i] Proposed by usjl[/i]

2002 Baltic Way, 14
Tags: geometry
Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.

1997 Dutch Mathematical Olympiad, 5
Tags: geometry , circumcenter , symmetry
Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$. a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.

2000 Turkey Team Selection Test, 3
Tags: number theory
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k. $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.

2019 Olympic Revenge, 3
Tags: geometry , incenter , euler line , construction
Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.

1986 IMO Shortlist, 14
Tags: geometry , incenter , circumcircle , collinearity
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2003 AMC 10, 3
Tags: induction
The sum of 5 consecutive even integers is $ 4$ less than the sum of the first $ 8$ consecutive odd counting numbers. What is the smallest of the even integers? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$

2022 Math Prize for Girls Problems, 1
Tags:
Determine the real value of $t$ that minimizes the expression \[ \sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}. \]

2006 MOP Homework, 3
Tags: vector , geometry
There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

1970 Putnam, B2
Tags: algebra , polynomial
The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

2006 AMC 8, 7
Tags: geometry
Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius. $ \textbf{(A)}\ X, Y, Z \qquad \textbf{(B)}\ Z, X, Y \qquad \textbf{(C)}\ Y, X, Z \qquad \textbf{(D)}\ Z, Y, X \qquad \textbf{(E)}\ X, Z, Y$

1955 AMC 12/AHSME, 33
Tags:
Henry starts a trip when the hands of the clock are together between $ 8$ a.m. and $ 9$ a.m. He arrives at his destination between $ 2$ p.m. and $ 3$ p.m. when the hands of the clock are exactly $ 180^\circ$ apart. The trip takes: $ \textbf{(A)}\ \text{6 hr.} \qquad \textbf{(B)}\ \text{6 hr. 43\minus{}7/11 min.} \qquad \textbf{(C)}\ \text{5 hr. 16\minus{}4/11 min.} \qquad \textbf{(D)}\ \text{6 hr. 30 min.} \qquad \textbf{(E)}\ \text{none of these}$

2019 PUMaC Algebra A, 2
Tags: algebra , relatively prime , number theory
Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.

2017 239 Open Mathematical Olympiad, 6
Tags: geometry , angle bisector , circumcircle
Given a circumscribed quadrilateral $ABCD$ in which $$\sqrt{2}(BC-BA)=AC.$$ Let $X$ be the midpoint of $AC$ and $Y$ a point on the angle bisector of $B$ such that $XD$ is the angle bisector of $BXY$. Prove that $BD$ is tangent to the circumcircle of $DXY$.

2021 MOAA, 7
Tags: speed
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$. $$xy+yz = 30$$ $$yz+zx = 36$$ $$zx+xy = 42$$ [i]Proposed by Nathan Xiong[/i]

1993 Bundeswettbewerb Mathematik, 1
Tags: integer , combinatorics , sum , summand
Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$

1983 All Soviet Union Mathematical Olympiad, 357
Tags: trigonometry
Two acute angles $a$ and $b$ satisfy condition $$\sin^2a+\sin^2b = \sin(a+b)$$ Prove that $a + b = \pi /2$.

2006 Germany Team Selection Test, 2
Tags: algebra , functional equation
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2019 Balkan MO Shortlist, G2
Tags: geometry
Let be a triangle $\triangle ABC$ with $m(\angle ABC) = 75^{\circ}$ and $m(\angle ACB) = 45^{\circ}$. The angle bisector of $\angle CAB$ intersects $CB$ at point $D$. We consider the point $E \in (AB)$, such that $DE = DC$. Let $P$ be the intersection of lines $AD$ and $CE$. Prove that $P$ is the midpoint of segment $AD$.

2002 Stanford Mathematics Tournament, 6
Tags: modular arithmetic
How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$?

2006 MOP Homework, 5
Tags: number theory
Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

2007 Germany Team Selection Test, 1
Tags: incenter , triangle , midpoint , congruent triangles , geometry
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.