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

2007 AMC 12/AHSME, 14

Tags: vieta
Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be distinct integers such that \[ (6 \minus{} a)(6 \minus{} b)(6 \minus{} c)(6 \minus{} d)(6 \minus{} e) \equal{} 45. \]What is $ a \plus{} b \plus{} c \plus{} d \plus{} e?$ $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

2019 India PRMO, 8

Tags: algebra
How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^{n}} + x + 1$ is divisible by $x^2 + x + 1$?

2015 Korea Junior Math Olympiad, 1

In an acute, scalene triangle $\triangle ABC$, let $O$ be the circumcenter. Let $M$ be the midpoint of $AC$. Let the perpendicular from $A$ to $BC$ be $D$. Let the circumcircle of $\triangle OAM$ hit $DM$ at $P(\not= M)$. Prove that $B, O, P$ are colinear.

1998 Israel National Olympiad, 7

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

2020 Miklós Schweitzer, 3

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

2002 Bundeswettbewerb Mathematik, 2

Tags: algebra
We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal. Prove that there is exactly a such sequence and find $a_{2002}$

2009 Puerto Rico Team Selection Test, 4

The point $ M$ is chosen inside parallelogram $ ABCD$. Show that $ \angle MAB$ is congruent to $ \angle MCB$, if and only if $ \angle MBA$ and $ \angle MDA$ are congruent.

2024 AIME, 1

Tags:
Every morning, Aya does a $9$ kilometer walk, and then finishes at the coffee shop. One day, she walks at $s$ kilometers per hour, and the walk takes $4$ hours, including $t$ minutes at the coffee shop. Another morning, she walks at $s+2$ kilometers per hour, and the walk takes $2$ hours and $24$ minutes, including $t$ minutes at the coffee shop. This morning, if she walks at $s+\frac12$ kilometers per hour, how many minutes will the walk take, including the $t$ minutes at the coffee shop?

1987 Putnam, A3

For all real $x$, the real-valued function $y=f(x)$ satisfies \[ y''-2y'+y=2e^x. \] (a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain. (b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.

2016 IFYM, Sozopol, 5

Tags: geometry
Points $K$ and $L$ are inner for $AB$ for an acute $\Delta ABC$, where $K$ is between $A$ and $L$. Let $P,Q$, and $H$ be the feet of the perpendiculars from $A$ to $CK$, from $B$ to $CL$, and from $C$ to $AB$, respectively. Point $M$ is the middle point of $AB$. If $PH\cap AC=X$ and $QH\cap BC=Y$, prove that points $H,P,M$, and $Q$ lie on one circle, if and only if the lines $AY,BX$, and $CH$ intersect in one point.

2016 Harvard-MIT Mathematics Tournament, 13

Tags:
A right triangle has side lengths $a$, $b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle.

2025 VJIMC, 2

Determine all real numbers $x>1$ such that \[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \] for any positive integer $n$.

2016 CMIMC, 4

Given a list $A$, let $f(A) = [A[0] + A[1], A[0] - A[1]]$. Alef makes two programs to compute $f(f(...(f(A))))$, where the function is composed $n$ times: \begin{tabular}{l|l} 1: \textbf{FUNCTION} $T_1(A, n)$ & 1: \textbf{FUNCTION} $T_2(A, n)$ \\ 2: $\quad$ \textbf{IF} $n = 0$ & 2: $\quad$ \textbf{IF} $n = 0$ \\ 3: $\quad$ $\quad$ \textbf{RETURN} $A$ & 3: $\quad$ $\quad$ \textbf{RETURN} $A$ \\ 4: $\quad$ \textbf{ELSE} & 4: $\quad$ \textbf{ELSE} \\ 5: $\quad$ $\quad$ \textbf{RETURN} $[T_1(A, n - 1)[0] + T_1(A, n - 1)[1],$ & 5: $\quad$ $\quad$ $B \leftarrow T_2(A, n - 1)$ \\ $\quad$ $\quad$ $\quad$ $T_1(A, n - 1)[0] - T_1(A, n - 1)[1]]$ & 6: $\quad$ $\quad$ \textbf{RETURN} $[B[0] + B[1], B[0] - B[1]]$ \\ \end{tabular} Each time $T_1$ or $T_2$ is called, Alef has to pay one dollar. How much money does he save by calling $T_2([13, 37], 4)$ instead of $T_1([13, 37], 4)$?

2016 Saudi Arabia BMO TST, 3

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

II Soros Olympiad 1995 - 96 (Russia), 11.5

The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?

2005 Romania National Olympiad, 1

Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$, have the form \[ f(z) = \alpha z + \beta \overline{z} \] for some complex $\alpha$, $\beta$. [i]Cristinel Mortici[/i]

2005 National Olympiad First Round, 3

Tags:
If the difference between the greatest and the smallest root of the equation $x^3 - 6x^2 + 5$ is equal to $F$, which of the following is true? $ \textbf{(A)}\ 0 \leq F < 2 \quad\textbf{(B)}\ 2 \leq F < 4 \quad\textbf{(C)}\ 4 \leq F < 6 \quad\textbf{(D)}\ 6 \leq F < 8 \quad\textbf{(E)}\ 8 \leq F $

2022 LMT Spring, 6

Tags: algebra
For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.

1979 Miklós Schweitzer, 1

Let the operation $ f$ of $ k$ variables defined on the set $ \{ 1,2,\ldots,n \}$ be called $ \textit{friendly}$ toward the binary relation $ \rho$ defined on the same set if \[ f(a_1,a_2,\ldots,a_k) \;\rho\ \;f(b_1,b_2,\ldots,b_k)\] implies $ a_i \; \rho \ b_i$ for at least one $ i,1\leq i \leq k$. Show that if the operation $ f$ is friendly toward the relations "equal to" and "less than," then it is friendly toward all binary relations. [i]B. Csakany[/i]

2016 Azerbaijan IMO TST First Round, 3

Tags: equation
Find the solution of the equation $8x(2x^2-1)(8x^4-8x^2+1)=1$ in the interval $[0,1]$?

2011 Baltic Way, 15

Tags: geometry
Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

1995 Dutch Mathematical Olympiad, 5

An array $ (a_1,a_2,...,a_{13})$ of $ 13$ integers is called $ tame$ if for each $ 1 \le i \le 13$ the following condition holds: If $ a_i$ is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called $ turbo$ $ tame$ if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum. $ (a)$ Give an example of a tame array of $ 13$ integers (not all equal). $ (b)$ Prove that in a tame array all numbers are of the same parity. $ (c)$ Prove that in a turbo tame array all numbers are equal.

2002 Romania Team Selection Test, 4

At an international conference there are four official languages. Any two participants can speak in one of these languages. Show that at least $60\%$ of the participants can speak the same language. [i]Mihai Baluna[/i]

1982 Putnam, A1

Tags:
Let $V$ be the region in the Cartesian plane consisting of all points $(x,y)$ satisfying the simultaneous conditions $$|x|\le y\le|x|+3\text{ and }y\le4.$$Find the centroid of $V$.

2025 Al-Khwarizmi IJMO, 5

Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime? [i]Marin Hristov, Bulgaria[/i]