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: 966

1953 Putnam, A2
Tags: Putnam , graph theory
The complete graph with 6 points and 15 edges has each edge colored red or blue. Show that we can find 3 points such that the 3 edges joining them are the same color.

1998 Putnam, 4
Tags: Putnam , floor function , college contests
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.\]

1998 Putnam, 5
Tags: Putnam , search , LaTeX , college contests
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.

1999 Putnam, 2
Tags: Putnam , algebra , polynomial , college contests
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]

2005 Putnam, A3
Tags: Putnam , algebra , polynomial , Stanford , college , function , geometry
Let $p(z)$ be a polynomial of degree $n,$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=\frac{p(z)}{z^{n/2}}.$ Show that all zeros of $g'(z)=0$ have absolute value $1.$

1970 Putnam, B2
Tags: Putnam , 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.

1996 Putnam, 4
Tags: Putnam , linear algebra , matrix , trigonometry , college contests
For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

1970 Putnam, B6
Tags: Putnam , geometry , area , quadrilateral
Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

1958 February Putnam, B6
Tags: Putnam , function , vector , integration , physics
A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.

2024 Putnam, B2
Tags: Putnam
Two convex quadrilaterals are called [i]partners[/i] if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent? [center][img]https://cdn.artofproblemsolving.com/attachments/6/e/cc9da12a49043410c50733cb6843e5ec1005d3.jpeg[/img][/center]

2011 Putnam, A2
Tags: Putnam , ratio , geometric series , college contests , Summation
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

1973 Putnam, A2
Tags: Putnam , Convergence , series
Consider an infinite series whose $n$-th term is $\pm (1\slash n)$, the $\pm$ signs being determined according to a pattern that repeats periodically in blocks of eight (there are $2^{8}$ possible patterns). (a) Show that a sufficient condition for the series to be conditionally convergent is that there are four "$+$" signs and four "$-$" signs in the block of eight signs. (b) Is this sufficient condition also necessary?

1980 Putnam, A1
Tags: Putnam , conics , parabola , algebra , polynomial , function
Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

2017 Putnam, A1
Tags: Putnam , Putnam 2017
Let $S$ be the smallest set of positive integers such that a) $2$ is in $S,$ b) $n$ is in $S$ whenever $n^2$ is in $S,$ and c) $(n+5)^2$ is in $S$ whenever $n$ is in $S.$ Which positive integers are not in $S?$ (The set $S$ is ``smallest" in the sense that $S$ is contained in any other such set.)

1991 Putnam, A3
Tags: Putnam , algebra , polynomial , college contests
Find all real polynomials $ p(x)$ of degree $ n \ge 2$ for which there exist real numbers $ r_1 < r_2 < ... < r_n$ such that (i) $ p(r_i) \equal{} 0, 1 \le i \le n$, and (ii) $ p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1$. [b]Follow-up:[/b] In terms of $ n$, what is the maximum value of $ k$ for which $ k$ consecutive real roots of a polynomial $ p(x)$ of degree $ n$ can have this property? (By "consecutive" I mean we order the real roots of $ p(x)$ and ignore the complex roots.) In particular, is $ k \equal{} n \minus{} 1$ possible for $ n \ge 3$?

1962 Putnam, B5
Tags: Putnam , inequalities
Prove that for every integer $n$ greater than $1:$ $$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$

Putnam 1938, B1
Tags: Putnam , linear algebra
Do either $(1)$ or $(2)$ $(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$. $(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$

1989 Putnam, A1
Tags: Putnam , algebra , difference of squares , special factorizations , college contests
How many base ten integers of the form 1010101...101 are prime?

1958 February Putnam, B1
Tags: Putnam , geometry , Triangles
i) Given line segments $A,B,C,D$ with $A$ the longest, construct a quadrilateral with these sides and with $A$ and $B$ parallel, when possible. ii) Given any acute-angled triangle $ABC$ and one altitude $AH$, select any point $D$ on $AH$, then draw $BD$ and extend until it intersects $AC$ in $E$, and draw $CD$ and extend until it intersects $AB$ in $F$. Prove that $\angle AHE = \angle AHF$.

2013 Putnam, 1
Tags: Putnam , college contests
For positive integers $n,$ let the numbers $c(n)$ be determined by the rules $c(1)=1,c(2n)=c(n),$ and $c(2n+1)=(-1)^nc(n).$ Find the value of \[\sum_{n=1}^{2013}c(n)c(n+2).\]

2015 Putnam, B6
Tags: Putnam , Putnam 2015 , Putnam number theory
For each positive integer $k,$ let $A(k)$ be the number of odd divisors of $k$ in the interval $\left[1,\sqrt{2k}\right).$ Evaluate: \[\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}k.\]

Putnam 1938, B6
Tags: Putnam
What is the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$ You may find it convenient to use the notation $h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.$ What is the algebraic condition for the plane not to intersect the ellipsoid?

2011 Putnam, A6
Tags: Putnam , abstract algebra , probability , limit , real analysis , linear algebra , matrix
Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$ Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.

Putnam 1939, B1
Tags: Putnam
The points $P(a,b)$ and $Q(0,c)$ are on the curve $\dfrac{y}{c} = \cosh{(\dfrac{x}{c})}.$ The line through $Q$ parallel to the normal at $P$ cuts the $x-$axis at $R.$ Prove that $QR = b.$

1957 Putnam, B6
Tags: Putnam , analytic geometry , graphing lines , differential equation
The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$-axis between $$x= \frac{3}{2} \pi \;\;\; \text{and} \;\;\; x= \sqrt{\frac{63}{53}} \pi.$$