К шапке сайта надо подключить поиск

window.onscroll = function showHeader() {
    var header = document.querySelector('.header');
    if(window.pageYOffset > 50){
        header.classList.add('header_fixed');
    } else{
        header.classList.remove('header_fixed');
    }
}

.container{
    width: 100%;
    max-width: 92%;
    margin: 0 auto;
}

.header-block{
    position: relative;
    z-index: 1;
    width: 100%;
    height: 75px;
    background: #006091;
}

.header-inner{
    display: flex;
    justify-content: space-between;
    align-items: center;
}

.header_fixed{
    position: fixed;
    z-index: 100;
    top: 0;
    left: 0;
    width: 100%;
    animation-name: showHeader;
    animation-duration: 1s;
}

@keyframes showHeader{
    0%
    {
        top: -70px;
        opacity: 0;
    }
    100%
    {
        top: 0;
        opacity: 1;
    }
}

.logo{
    width: 210px;
    height: 75px;
    background: url(../images/Logo.png);
    -webkit-background-size: cover;
            background-size: cover;
}

.logo a{
	z-index: 100;
	width: 210px;
    height: 75px;
}



.nav{
    font-size: 20px;
}

.nav-link{
    position: relative;
    display: inline-block;
    margin: 0 15px;
    padding-left: 10px;
    vertical-align: top;
    color: #fff;
    border-left: 2px solid #fff;
}

.nav-link a:hover {
	transition: all .2s ease;
  color: #f2b46d;
}

.down, .up{
	font-size: 13px;
}

.submenu{
	  width: 685px;
    height: 220px;
    margin-top: 80px;
    background: #888;
    opacity: 0.8;
    display: none;
}

.submenu ul li {
    float: none;
    list-style: none;
    text-align: center;
    margin-bottom: 5px;
}
.submenu ul li a {
    display: block;
    width: 135px;
    height: 20px;
    text-decoration: none;
    font-size: 17px;
    color: white;
    cursor: pointer;
    padding-top: 3px;
}

<header class="header">
	<div class="header-block">
	<div class="container">
		<div class="header-inner">
			<a href="#">
			<div class="logo">
			</div>
			</a>
			 <nav class="nav">
			 	<ul class="menu">
			  <li class="nav-link"><a href="/images/page/a1.jpg">Абитуриенту
			  	<span class="down">	&#9660;</span>
			  	<span class="up">	&#9650;</span></a>
			  	<ul class="submenu">
			  		<li><a href="#">Перечень необходимых документов</a></li>
			  	</ul>
			  </li>
			  <li class="nav-link"><a href="/images/page/01.jpg">Обучение
			    <span class="down">	&#9660;</span>
			  	<span class="up">	&#9650;</span></a>
			  <ul class="submenu">
			  		<li><a href="#">Бакалавриат</a></li>
			  		<li><a href="#">Колледж</a></li>
			  		<li><a href="#">Курсы</a></li>
			  	</ul>
			  </li>
			  <li class="nav-link"><a href="/images/page/n1.jpg">Новости</a></li>
			  <li class="nav-link"><a href="/images/page/o1.jpg">О нас</a></li>
			  </ul>
			 </nav>
		 </div>
	</div>
	</div>
</header>